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:''The following discussion is an archived debate of the proposed deletion of the article below. <span style="color:red">'''Please do not modify it.'''</span> Subsequent comments should be made on the appropriate discussion page (such as the article's talk page or in a ]). No further edits should be made to this page. ''
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The result of the debate was '''no consensus'''. Those supporting a merge can slap {{tl|merge}} on the article and thrash this out on the talk. Right now there is no consensus to delete, so no decision has been made. Please work this dispute out on the talk, and please '''do not''' equate "no consensus" with "keep". If you must oppose a merge/redirect/whatever, please do not cite the "no consensus" decision alone to back up your case. ] | ] 16:12, 12 March 2006 (UTC)

===]=== ===]===
''I have refactored comments from this page to ] to improve readability and reduce the amount of space people viewing the entire day have to scroll through. Please use the talk page in future for long discussion. This should not be taken as implying those comments are less valuable in any way, and you are urged to read both this and the talk page.'' ] 09:54, 9 March 2006 (UTC)
: Logically, shouldn't that be moved to the talk page of the article, instead of the delete discussion? &mdash; ] | ] 13:53, 9 March 2006 (UTC)

This redundant article serves no purpose. (1) and (2) are covered by ]; (3) is covered by ]. These relationships are already explored in detail at ]. In 30 months, the article has accumulated as many edits; I attempted to merge it but was reverted. ] 01:44, 6 March 2006 (UTC) This redundant article serves no purpose. (1) and (2) are covered by ]; (3) is covered by ]. These relationships are already explored in detail at ]. In 30 months, the article has accumulated as many edits; I attempted to merge it but was reverted. ] 01:44, 6 March 2006 (UTC)
*'''Keep.''' Melchoir's hostility seems gratuitous; I don't know where it comes from. His attempt to "merge" material into ] amounted to (1) paraphrasing a fragment of this article in a way that made clear that he understood none of it and didn't care to; and (2) putting it into a randomly chosen place in that article. ] 01:49, 6 March 2006 (UTC) *'''Keep.''' Melchoir's hostility seems gratuitous; I don't know where it comes from. His attempt to "merge" material into ] amounted to (1) paraphrasing a fragment of this article in a way that made clear that he understood none of it and didn't care to; and (2) putting it into a randomly chosen place in that article. ] 01:49, 6 March 2006 (UTC)
**I do not understand why it is necessary to insult me to this degree. I have made nothing but good-faith edits; I understand the articles a little better than not at all, and I make sure that I understand what I'm doing before I do it. You can't possibly think my choice of position was random. And I merged a "fragment" because there is exactly one sentence in this article that can't be found in a more relevant place. ] 02:07, 6 March 2006 (UTC) **I do not understand why it is necessary to insult me to this degree. I have made nothing but good-faith edits; I understand the articles a little better than not at all, and I make sure that I understand what I'm doing before I do it. You can't possibly think my choice of position was random. And I merged a "fragment" because there is exactly one sentence in this article that can't be found in a more relevant place. ] 02:07, 6 March 2006 (UTC)
***Michael Hardy's hostility seems gratuitous. But what do I know? I'm an anon posting from a shared IP. --] 14:00, 9 March 2006 (UTC)
***They did not look at all like good-faith edits. I did not insult; I accused. And your writings above on this votes-for-deletion page clearly show that you have no understanding of this article. You say that (1) is covered somewhere and (2) is covered somewhere, and (3) can also be covered somewhere, and that misses the point. This is not about three disparate topics, each of which should be covered somewhere; it's about a '''triad''' and the relationship---in particular the contrast and the commonality---among the three things. You write "there is exactly one sentence in this article that can't be found in a more relevant place"; that makes it clear that you think '''each''' of the three things should be covered '''somewhere''', possibly separately, so that you've missed the point that it's about a relationship. And yes, it does appear that you put it in a random place. The place where you put it within the article is utterly inappropriate and very stupidly so in a way that shows reckless disregard for what part of the article it is in. It also fails to convey the information. The subsection into which you inserted it was devoted to pointing out that the starting value could be ''any'' integer. Why would you put it there, of all places? And you began by saying "Such a strategy can also be useful to prove a statement for all ''n''. That's what the ''whole'' article is about, unless by "all ''n''" you mean something other than all ''n'' in a sequence with a first element, a second, a third, and so on. "Such a strategy"?? "Useful"?? The ''point'' was the splitting into three. Where is that in what you wrote? That small fragment would clearly fit better into the section on transfinite induction. But that would mislead: this particular form is obviously not used '''only''' in transfinite induction. Moreover, the sentence you wrote is incomprehensible; it is impossible to tell whether there is a "then" at any of the various semicolons you put into the sentence. I invite any mathematician here to look at Melchoir's edit at and see whether it conveys any of the meaning. Don't just look at what he wrote; look at whether it's in an appropriate place by reading the short section into which he inserted it. ] 02:31, 6 March 2006 (UTC)
****] is the place to explore relationships between types of mathematical induction, and it already does. The semicolons are not mine, and if you think you can improve the wording, you should try it. Finally, you continue to insult me with such accusations as "very stupidly". This is not constructive; please stop. ] 02:39, 6 March 2006 (UTC)
***] does not explain the contrast between these three forms. It mentions two of the three (#1 and #3, but not #2) but not in a way that calls explicit attention to the contrast between the three. It gives only two exmample (and that's one thing that this present article needs). One of the two is somewhat deficient in that only a small part of the induction hypothesis is used and the example lacks detail. The very fact that more examples, with more detail, should be added to ], is one reason why this article should be separate from it: so that the two sorts of discussion will not interfere with each other (when you're paying attention to either of them, the other becomes noise). The examples that need to be added to the present article should be done in a different way from the examples in ]: the emphasis in examples in ''this'' should be on something other than details of the proofs. The edit history does make it look as if the semicolons were Melchoir's. Melchoir, I naturally presume that anyone who takes an interest in these topics is capable of judiciously choosing an appropriate place in the article and otherwise understanding and writing clearly; your failure to do any of those was so complete that it looked like recklessness rather than lack of any ability; that is why I accused you of that. It is of course possible that my accusation was incorrect, but saying you did something stupidly when you had the ability to do otherwise is an accusation, rather than an insult. ] 03:25, 6 March 2006 (UTC)
****I believe it is possible, reasonable, necessary, and consistent with the existing article to contrast forms of mathematical induction in ]. The incompleteness of ] is unfortunate, and ] does not help.
:::As for "...understanding and writing clearly; your failure to do any of those was so complete...", perhaps I need to quote ] here: "Comment on content, not on the contributor". My writing skills are not on trial here. And I am not aware of a definition of "insult" that excludes accusations of stupidity. Please focus on the articles. ] 03:41, 6 March 2006 (UTC)
*'''Keep''' It's also cool having the 3 types right there clean and consisely ] 01:51, 6 March 2006 (UTC) *'''Keep''' It's also cool having the 3 types right there clean and consisely ] 01:51, 6 March 2006 (UTC)
*'''Delete''' per Melchoir. --] 02:11, 6 March 2006 (UTC) *'''Delete''' per Melchoir. --] 02:11, 6 March 2006 (UTC)
*'''Merge''' any original information. ]]]<nowiki>ash</nowiki>] ] 02:13, 6 March 2006 (UTC) *<s>'''Merge''' any original information. ]]]<nowiki>ash</nowiki>] ] 02:13, 6 March 2006 (UTC)</s> '''Keep''' ]]]<nowiki>ash</nowiki>] ] 00:04, 7 March 2006 (UTC)
*'''Merge''' per nom. No clear reason for this topic to have its own article. ]] 03:37, 6 March 2006 (UTC). *'''Merge''' per nom. No clear reason for this topic to have its own article. ]] 03:37, 6 March 2006 (UTC).
**Or '''delete''', if the content isn't worth preserving. (I share ]'s concerns.) ]] 05:37, 6 March 2006 (UTC) **Or '''delete''', if the content isn't worth preserving. (I share ]'s concerns.) ]] 05:37, 6 March 2006 (UTC)
*'''Delete''' essentially per nom, as I don't see anything in here to merge. This may be the first example of mathcruft I've seen. How is (2) functionally different from (1)? Whatever your answer is, then please articulate why there should be the explicit case where one has to prove ''n=1,2'' by hand and then start the induction at 2? And etc.? In any case, this article contains absolutely zero mathematical content not contained in other articles. --] (]) 05:17, 6 March 2006 (UTC) *'''Delete''' essentially per nom, as I don't see anything in here to merge. This may be the first example of mathcruft I've seen. How is (2) functionally different from (1)? Whatever your answer is, then please articulate why there should be the explicit case where one has to prove ''n=1,2'' by hand and then start the induction at 2? And etc.? In any case, this article contains absolutely zero mathematical content not contained in other articles. --] (]) 05:17, 6 March 2006 (UTC)
**If you see no difference between (1) and (2), then you're not paying attention. "Why there '''SHOULD''' be"? I never said there '''"SHOULD"''' be; I said there '''ARE''' very many such cases. I suggest if you don't know that, you're simply not a mathematician. ] 21:50, 6 March 2006 (UTC)
⚫ *'''Delete''', content isn't worth preserving. --] 06:31, 6 March 2006 (UTC)
*'''Delete''', content isn't worth preserving. It's just hairsplitting over the initial step. Maths undergrad that DOES understand the material fully and still sees no point in it.--] 06:31, 6 March 2006 (UTC)
*'''Delete'''; no offense to Michael, but this just doesn't strike me as an encyclopedia article. It would be a good sort of observation to include in a textbook, maybe for a discrete math course. --] 06:55, 6 March 2006 (UTC)
*<s>'''Delete''' as per ]. If the article's originator can work up a '''merge''', then please go for it.</s>] 12:13, 6 March 2006 (UTC) Withdrawing my vote. I don't have the expertise to comment on this issue. (BTW, thanks for putting the message on my talk page, I wouldn't have come back here otherwise).] 09:59, 7 March 2006 (UTC)
*'''Delete'''. Comparison of different variations on induction belongs at ]. This article seems like it might be about using induction for a certain group of problems. If this topic really does deserve an article, it probably needs a better title, and it should at least have an introduction which makes the topic clear. ] (]) 12:17, 6 March 2006 (UTC)
*'''Merge and Delete'''. Whatever new content here should be merged and this article deleted. -- ] 14:39, 6 March 2006 (UTC)
**'''Comment''' I don't think you can really do that, in general, because of the GFDL. Certainly we wouldn't want to just stick literal text from the article somewhere else and then delete the record of who wrote it. Possibly if the text were paraphrased first we'd be technically OK; I'm really not sure what the rules on that are. --] 15:17, 6 March 2006 (UTC)
***'''Comment''': This is correct. A merge must be followed with a redirect. ] 09:54, 9 March 2006 (UTC)
*'''Keep''' per MadCow257. --<span style="background:gold;">]]</span>] 14:52, 6 March 2006 (UTC)
*'''Delete''', as per nom. The ] article is totally superior in content, grammar, comprehensibility and every other way imaginable. This article is badly written, and contains no new, or interesting, information. It's hardly surprising the nominator "didn't understand" the new article. It's borderline gibberish. ]
*'''Delete''' No material seems salvagable for merger. ] 16:16, 6 March 2006 (UTC)
⚫ *'''Delete'''- there is nothing here that isn't covered elsewhere. ] 21:11, 6 March 2006 (UTC)
<s>*'''Delete''' per well argued nomination. ]]]] 21:54, 6 March 2006 (UTC)</s>
::'''Keep.''' ]]]] 23:24, 6 March 2006 (UTC)
<s>*'''Delete''' per nom. ''Possible'' '''merge''' if someone is up to the task. No question that ] is written to a higher standard. I also suggest that ] find another supporter from the academic math community to back his assertion that the "article does indeed contain information not in that other article". </s>] 22:43, 6 March 2006 (UTC)
*'''Keep''' provided a source can be cited, per ]. Otherwise, still '''delete'''. Appreciate the perceived good faith edits of the author, so I have conditionally changed my vote. ] 23:52, 6 March 2006 (UTC)
*'''Delete'''. The article is not bad and is understandable, but I am concerned about ]. I've never heard the term "three forms of mathematical induction". --]] 04:35, 7 March 2006 (UTC)

*'''Delete'''. Of no value as an independent topic --] 23:29, 8 March 2006 (UTC)

*'''Merge'''. This page is really about what it itself calls the "second form" of induction, which isn't really dealt with on the ] page. This is an interesting topic and completely merits ]'s expanded content, but there's no reason for it to be anything other than a subhead in ] just as the "first" and "third" forms currently are. —] 03:03, 7 March 2006 (UTC)

*'''Delete'''. I would say that the focus of the article is to present a trichotomy of techniques, and give examples to illustrate the distinctions among them. However, I am not convinced that either the trichotomy, at least as it is presented here, really exists, and I do not think that the examples illustrate anything substantial. Furthermore, as it's been noted, the factual material that's contained here is present in the ] article, so that the sole original content of the article becomes the uninformative examples. Let me elaborate: <small>sent to talk page by ]</small> ] 04:19, 7 March 2006 (UTC)
*'''Keep''' and '''Restructure''' - the point is actually about a common difficulty in using induction (2). So it could be rewritten into an article about Polya's example (which is quite famous), introducing the stuff on 2nd form to explain the apparent paradox. The discussion of 1 and 3 could be probably merged into the ] article. ] 08:10, 7 March 2006 (UTC)

*<s>'''merge'''</s>. Form 2 can be easily be rendered as form 1, by a simple renumbering. Define two sequences of cases: let <math>case_A</math> the original cases and let <math>case_B(n)= case_A(n+1)</math>, be a renumbered sequence. So the form 2 induction for <math>case_A</math> just becomes a form 1 induction for <math>case_B</math>, with an extra vaciously true case <math>case_B(0)</math>. In general induction arguments don't have to start at 1, if you can prove all cases up to ''m'' by other means and then prove &gt;''m'' using induction then its just as good an argument. Its a shame we've had to go here, instead of reverting the merge notice, which imediatly led to and afd in retaliation, a discussion on the merge would have been more civil.--] (]) 13:22, 7 March 2006 (UTC)
**'''rename''' I now think that this article is missnamed. The key points seem to be more about inductive or recirsive arguments relating to (binary) operations. It does not use the normal definition of induction, interms of a sequence of statements, but instead requires extra conditions linking the statement number to the number of operands.--] (]) 00:38, 10 March 2006 (UTC)
*'''delete''' as the selection of the three forms is ]. (This is an oddity, as I see it. All of the article could be merged into different articles.) My suggestion would be to find a good name for the 2nd form, and keep that as a separate article, merging all the rest into ]. Specifically -- forms 1 and 2 are the same '''type''' of induction, with the difficulties in different steps, while form 3 is '''different'''. They should not all be in the same article without including a number of of still different forms. Furthermore, the triangle inquality miscounts "n". The step that is impossible is '''2''' to '''3''', rather than '''1''' to '''2''', as the trivial equation is <math>d \left( x_1,x_1 \right) \leq 0</math>. (In other words, I agree with ], but feel, in addition, that the selection of items in the article is ].) &mdash; ] | ] 15:03, 7 March 2006 (UTC)
*'''Keep''', possibly '''merge''' existing text into ] (if it is placed after the explanation of mathematical induction, the necessary level of marthematical competence may be assumed). But this is a question of ordinary editing. No claim made here justifies deletion. ] 16:18, 7 March 2006 (UTC)
*'''Merge''' Polya's example and perhaps the product rule example into ] or some article on logical ] or erroneous ]. Delete the rest, as it is redundant or too detailed (maybe put into a Wikibook on proof techniques?). This article is too basic to appeal to readers who fully understand induction already. I also agree about the ] problem; I have never seen this classification before, there are no sources, and there is no reason given to single out these three forms of induction from others possible. ] 17:48, 7 March 2006 (UTC)
**The essence of Polya's example is '''not erroneous proof'''; the essense is the '''form''' of the argument. What is erroneous is there only because Polya wanted to divorce the form from specific cases, and also to set an exercise for students. But Polya's example illustrates the form perfectly. Those who think it's erroneous to say all horses are of the same color are being too literal-minded; sometimes all horses are of the same color. Don't construe that literally. ] 21:23, 7 March 2006 (UTC)
*'''Partial merge, userfy and potentially move to wikibooks'''. Article presents an interesting and to me somewhat new perspective on mathematical induction (bear in mind, I have not been an active mathematician for 5 years). This perspective should be integrated into the article on the topic at hand, namely mathematical induction, to the extent appropriate. This will of course involve discussion and debate with the denizens of that article. Beyond this, could become part of a mathematical wikibook of some sort; pending that, should be userfied. ] 04:23, 9 March 2006 (UTC)
*'''Comment'''. Would be interesting to hear the perspective of active participants in Wikiproject Mathematics, not as much on content or to give them any sort of veto power, but to understand what has worked in the past regarding interesting perspectives on one aspect of an encyclopedic mathematical topic. Heck, I have all sorts of material from old lectures I gave where I presented something mathematical from a slightly iconoclastic angle, but which is not new research. What is the right approach to judge what belongs on Misplaced Pages? ] 04:23, 9 March 2006 (UTC)

*'''Transwiki''' to Wikibooks. Misplaced Pages is not a how-to, and articles need to be comprehensible to people outside a narrow field of competence. Not making any assesment about the article here. ] 09:48, 9 March 2006 (UTC)
**This has certainly never been a how-to article in any of its versions. And I think it is comprehensible to all mathematicians, not just to those in any narrow field of competence. Should all Misplaced Pages articles that are incomprehensible to non-mathematicians get deleted? What is that -- maybe 10% of all Misplaced Pages articles, and 90% of the ones on mathematics? ] 22:31, 9 March 2006 (UTC)

*'''delete''' The point of this article is completely lost to me, if it even has a point. I don't see anything significant in the distinctions made. If Polya's example is famous it should be merged to the main induction article. -] 13:25, 10 March 2006 (UTC)

*'''delete''' (or maybe '''merge''') Perhaps something in this article belong in ], but other than that it seems to be a bunch of howto and examples, and gets hung up on the point of whether the base case of an induction is vacuous (which might deserve one sentence in the induction article). ] 00:08, 12 March 2006 (UTC)
**How in the world does anyone manage to see this as a "how to"?? It skips details precisely because it is NOT a how-to. There is NO mention of whether the base case is vacuous in the examples, but rather only in the case of complete induction, where it explains why the base case is always vacuous. In all three examples, the base case is asserted to be the substantial part--as far from vacuous as you can get. ] 03:17, 12 March 2006 (UTC)


:''The above discussion is preserved as an archive of the debate. <span style="color:red">'''Please do not modify it.'''</span> Subsequent comments should be made on the appropriate discussion page (such as the article's talk page or in a ]). No further edits should be made to this page.''</div>

Latest revision as of 02:28, 12 February 2023

The following discussion is an archived debate of the proposed deletion of the article below. Please do not modify it. Subsequent comments should be made on the appropriate discussion page (such as the article's talk page or in a deletion review). No further edits should be made to this page.

The result of the debate was no consensus. Those supporting a merge can slap {{merge}} on the article and thrash this out on the talk. Right now there is no consensus to delete, so no decision has been made. Please work this dispute out on the talk, and please do not equate "no consensus" with "keep". If you must oppose a merge/redirect/whatever, please do not cite the "no consensus" decision alone to back up your case. Johnleemk | Talk 16:12, 12 March 2006 (UTC)

Three forms of mathematical induction

I have refactored comments from this page to Misplaced Pages talk:Articles for deletion/Three forms of mathematical induction to improve readability and reduce the amount of space people viewing the entire day have to scroll through. Please use the talk page in future for long discussion. This should not be taken as implying those comments are less valuable in any way, and you are urged to read both this and the talk page. Stifle 09:54, 9 March 2006 (UTC)

Logically, shouldn't that be moved to the talk page of the article, instead of the delete discussion? — Arthur Rubin | (talk) 13:53, 9 March 2006 (UTC)

This redundant article serves no purpose. (1) and (2) are covered by Mathematical induction; (3) is covered by complete induction. These relationships are already explored in detail at Mathematical induction. In 30 months, the article has accumulated as many edits; I attempted to merge it but was reverted. Melchoir 01:44, 6 March 2006 (UTC)

  • Keep. Melchoir's hostility seems gratuitous; I don't know where it comes from. His attempt to "merge" material into mathematical induction amounted to (1) paraphrasing a fragment of this article in a way that made clear that he understood none of it and didn't care to; and (2) putting it into a randomly chosen place in that article. Michael Hardy 01:49, 6 March 2006 (UTC)
    • I do not understand why it is necessary to insult me to this degree. I have made nothing but good-faith edits; I understand the articles a little better than not at all, and I make sure that I understand what I'm doing before I do it. You can't possibly think my choice of position was random. And I merged a "fragment" because there is exactly one sentence in this article that can't be found in a more relevant place. Melchoir 02:07, 6 March 2006 (UTC)
  • Keep It's also cool having the 3 types right there clean and consisely MadCow257 01:51, 6 March 2006 (UTC)
  • Delete per Melchoir. --Allen 02:11, 6 March 2006 (UTC)
  • Merge any original information. Royboycrashfan 02:13, 6 March 2006 (UTC) Keep Royboycrashfan 00:04, 7 March 2006 (UTC)
  • Merge per nom. No clear reason for this topic to have its own article. dbtfz 03:37, 6 March 2006 (UTC).
  • Delete essentially per nom, as I don't see anything in here to merge. This may be the first example of mathcruft I've seen. How is (2) functionally different from (1)? Whatever your answer is, then please articulate why there should be the explicit case where one has to prove n=1,2 by hand and then start the induction at 2? And etc.? In any case, this article contains absolutely zero mathematical content not contained in other articles. --Deville (Talk) 05:17, 6 March 2006 (UTC)
    • If you see no difference between (1) and (2), then you're not paying attention. "Why there SHOULD be"? I never said there "SHOULD" be; I said there ARE very many such cases. I suggest if you don't know that, you're simply not a mathematician. Michael Hardy 21:50, 6 March 2006 (UTC)
  • Delete, content isn't worth preserving. It's just hairsplitting over the initial step. Maths undergrad that DOES understand the material fully and still sees no point in it.--Mmx1 06:31, 6 March 2006 (UTC)
  • Delete; no offense to Michael, but this just doesn't strike me as an encyclopedia article. It would be a good sort of observation to include in a textbook, maybe for a discrete math course. --Trovatore 06:55, 6 March 2006 (UTC)
  • Delete as per Melchoir. If the article's originator can work up a merge, then please go for it.Vizjim 12:13, 6 March 2006 (UTC) Withdrawing my vote. I don't have the expertise to comment on this issue. (BTW, thanks for putting the message on my talk page, I wouldn't have come back here otherwise).Vizjim 09:59, 7 March 2006 (UTC)
  • Delete. Comparison of different variations on induction belongs at mathematical induction. This article seems like it might be about using induction for a certain group of problems. If this topic really does deserve an article, it probably needs a better title, and it should at least have an introduction which makes the topic clear. JPD (talk) 12:17, 6 March 2006 (UTC)
  • Merge and Delete. Whatever new content here should be merged and this article deleted. -- Alpha269 14:39, 6 March 2006 (UTC)
    • Comment I don't think you can really do that, in general, because of the GFDL. Certainly we wouldn't want to just stick literal text from the article somewhere else and then delete the record of who wrote it. Possibly if the text were paraphrased first we'd be technically OK; I'm really not sure what the rules on that are. --Trovatore 15:17, 6 March 2006 (UTC)
  • Keep per MadCow257. --Siva1979 14:52, 6 March 2006 (UTC)
  • Delete, as per nom. The mathematical induction article is totally superior in content, grammar, comprehensibility and every other way imaginable. This article is badly written, and contains no new, or interesting, information. It's hardly surprising the nominator "didn't understand" the new article. It's borderline gibberish. GWO
  • Delete No material seems salvagable for merger. JoshuaZ 16:16, 6 March 2006 (UTC)
  • Delete- there is nothing here that isn't covered elsewhere. Reyk 21:11, 6 March 2006 (UTC)

*Delete per well argued nomination. εγκυκλοπαίδεια* 21:54, 6 March 2006 (UTC)

Keep. εγκυκλοπαίδεια* 23:24, 6 March 2006 (UTC)

*Delete per nom. Possible merge if someone is up to the task. No question that mathematical induction is written to a higher standard. I also suggest that Michael find another supporter from the academic math community to back his assertion that the "article does indeed contain information not in that other article". Slowmover 22:43, 6 March 2006 (UTC)

  • Keep provided a source can be cited, per Allen. Otherwise, still delete. Appreciate the perceived good faith edits of the author, so I have conditionally changed my vote. Slowmover 23:52, 6 March 2006 (UTC)
  • Delete. The article is not bad and is understandable, but I am concerned about WP:NOR. I've never heard the term "three forms of mathematical induction". --C S (Talk) 04:35, 7 March 2006 (UTC)
  • Delete. I would say that the focus of the article is to present a trichotomy of techniques, and give examples to illustrate the distinctions among them. However, I am not convinced that either the trichotomy, at least as it is presented here, really exists, and I do not think that the examples illustrate anything substantial. Furthermore, as it's been noted, the factual material that's contained here is present in the mathematical induction article, so that the sole original content of the article becomes the uninformative examples. Let me elaborate: sent to talk page by Stifle Ryan Reich 04:19, 7 March 2006 (UTC)
  • Keep and Restructure - the point is actually about a common difficulty in using induction (2). So it could be rewritten into an article about Polya's example (which is quite famous), introducing the stuff on 2nd form to explain the apparent paradox. The discussion of 1 and 3 could be probably merged into the Mathematical induction article. AdamSmithee 08:10, 7 March 2006 (UTC)
  • merge. Form 2 can be easily be rendered as form 1, by a simple renumbering. Define two sequences of cases: let c a s e A {\displaystyle case_{A}} the original cases and let c a s e B ( n ) = c a s e A ( n + 1 ) {\displaystyle case_{B}(n)=case_{A}(n+1)} , be a renumbered sequence. So the form 2 induction for c a s e A {\displaystyle case_{A}} just becomes a form 1 induction for c a s e B {\displaystyle case_{B}} , with an extra vaciously true case c a s e B ( 0 ) {\displaystyle case_{B}(0)} . In general induction arguments don't have to start at 1, if you can prove all cases up to m by other means and then prove >m using induction then its just as good an argument. Its a shame we've had to go here, instead of reverting the merge notice, which imediatly led to and afd in retaliation, a discussion on the merge would have been more civil.--Salix alba (talk) 13:22, 7 March 2006 (UTC)
    • rename I now think that this article is missnamed. The key points seem to be more about inductive or recirsive arguments relating to (binary) operations. It does not use the normal definition of induction, interms of a sequence of statements, but instead requires extra conditions linking the statement number to the number of operands.--Salix alba (talk) 00:38, 10 March 2006 (UTC)
  • delete as the selection of the three forms is WP:OR. (This is an oddity, as I see it. All of the article could be merged into different articles.) My suggestion would be to find a good name for the 2nd form, and keep that as a separate article, merging all the rest into Mathematical induction. Specifically -- forms 1 and 2 are the same type of induction, with the difficulties in different steps, while form 3 is different. They should not all be in the same article without including a number of of still different forms. Furthermore, the triangle inquality miscounts "n". The step that is impossible is 2 to 3, rather than 1 to 2, as the trivial equation is d ( x 1 , x 1 ) ≤ 0 {\displaystyle d\left(x_{1},x_{1}\right)\leq 0} . (In other words, I agree with AdamSmithee, but feel, in addition, that the selection of items in the article is WP:OR.) — Arthur Rubin | (talk) 15:03, 7 March 2006 (UTC)
  • Keep, possibly merge existing text into mathematical induction (if it is placed after the explanation of mathematical induction, the necessary level of marthematical competence may be assumed). But this is a question of ordinary editing. No claim made here justifies deletion. Septentrionalis 16:18, 7 March 2006 (UTC)
  • Merge Polya's example and perhaps the product rule example into mathematical induction or some article on logical fallacy or erroneous proof. Delete the rest, as it is redundant or too detailed (maybe put into a Wikibook on proof techniques?). This article is too basic to appeal to readers who fully understand induction already. I also agree about the WP:OR problem; I have never seen this classification before, there are no sources, and there is no reason given to single out these three forms of induction from others possible. Joshuardavis 17:48, 7 March 2006 (UTC)
    • The essence of Polya's example is not erroneous proof; the essense is the form of the argument. What is erroneous is there only because Polya wanted to divorce the form from specific cases, and also to set an exercise for students. But Polya's example illustrates the form perfectly. Those who think it's erroneous to say all horses are of the same color are being too literal-minded; sometimes all horses are of the same color. Don't construe that literally. Michael Hardy 21:23, 7 March 2006 (UTC)
  • Partial merge, userfy and potentially move to wikibooks. Article presents an interesting and to me somewhat new perspective on mathematical induction (bear in mind, I have not been an active mathematician for 5 years). This perspective should be integrated into the article on the topic at hand, namely mathematical induction, to the extent appropriate. This will of course involve discussion and debate with the denizens of that article. Beyond this, could become part of a mathematical wikibook of some sort; pending that, should be userfied. Martinp 04:23, 9 March 2006 (UTC)
  • Comment. Would be interesting to hear the perspective of active participants in Wikiproject Mathematics, not as much on content or to give them any sort of veto power, but to understand what has worked in the past regarding interesting perspectives on one aspect of an encyclopedic mathematical topic. Heck, I have all sorts of material from old lectures I gave where I presented something mathematical from a slightly iconoclastic angle, but which is not new research. What is the right approach to judge what belongs on Misplaced Pages? Martinp 04:23, 9 March 2006 (UTC)
  • Transwiki to Wikibooks. Misplaced Pages is not a how-to, and articles need to be comprehensible to people outside a narrow field of competence. Not making any assesment about the article here. Stifle 09:48, 9 March 2006 (UTC)
    • This has certainly never been a how-to article in any of its versions. And I think it is comprehensible to all mathematicians, not just to those in any narrow field of competence. Should all Misplaced Pages articles that are incomprehensible to non-mathematicians get deleted? What is that -- maybe 10% of all Misplaced Pages articles, and 90% of the ones on mathematics? Michael Hardy 22:31, 9 March 2006 (UTC)
  • delete The point of this article is completely lost to me, if it even has a point. I don't see anything significant in the distinctions made. If Polya's example is famous it should be merged to the main induction article. -MarSch 13:25, 10 March 2006 (UTC)
  • delete (or maybe merge) Perhaps something in this article belong in Complete induction, but other than that it seems to be a bunch of howto and examples, and gets hung up on the point of whether the base case of an induction is vacuous (which might deserve one sentence in the induction article). Rdore 00:08, 12 March 2006 (UTC)
    • How in the world does anyone manage to see this as a "how to"?? It skips details precisely because it is NOT a how-to. There is NO mention of whether the base case is vacuous in the examples, but rather only in the case of complete induction, where it explains why the base case is always vacuous. In all three examples, the base case is asserted to be the substantial part--as far from vacuous as you can get. Michael Hardy 03:17, 12 March 2006 (UTC)


The above discussion is preserved as an archive of the debate. Please do not modify it. Subsequent comments should be made on the appropriate discussion page (such as the article's talk page or in a deletion review). No further edits should be made to this page.