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This is an old revision of this page, as edited by C S (talk | contribs) at 23:48, 7 March 2006 (→[[Three forms of mathematical induction]]: response to Michael). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

Three forms of mathematical induction

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)[reply]

  • 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)[reply]
    • 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)[reply]
      • 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 [1] 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. Michael Hardy 02:31, 6 March 2006 (UTC)[reply]
        • Mathematical induction 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. Melchoir 02:39, 6 March 2006 (UTC)[reply]
      • Mathematical induction 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 mathematical induction, 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 mathematical induction: 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. Michael Hardy 03:25, 6 March 2006 (UTC)[reply]
As for "...understanding and writing clearly; your failure to do any of those was so complete...", perhaps I need to quote Wikipedia:No personal attacks 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. Melchoir 03:41, 6 March 2006 (UTC)[reply]
0 is not a sum of fewer than 0 numbers each of which is smaller than 0;
1 is not a sum of fewer than 1 numbers each of which is smaller than 1;
2 is not a sum of fewer than 2 numbers each of which is smaller than 2;
3 IS a sum of fewer than 3 numbers each of which is smaller than 3;
generally, n IS a sum of fewer than n numbers each of which is smaller than n, for n ≥ 3.
Comment on the relevance to this article. The numbers 0, 1, and 2 correspond to the three forms. Michael Hardy 21:54, 6 March 2006 (UTC)[reply]
      • There are many variants of induction. All you have in these two cases is really a difference between what step is the most difficult. It is still functionally the same procedure. So this really isn't a difference that matters much at all. JoshuaZ 21:55, 6 March 2006 (UTC)[reply]
This is entirely mistaken. The form of the argument is different. Look closely at the examples. Michael Hardy 01:37, 7 March 2006 (UTC)[reply]
Can't this be mentioned at Mathematical_induction#Start_at_b as another example with b = 3? Melchoir 22:24, 6 March 2006 (UTC)[reply]
No, because that's not what this is at all; I never even thought of using mathematical induction to prove the above. The point of the above is that it explains why there are these three forms, corresponding to 0, 1, and 2 (or actually 1, 2, and 0, in that order, that being the order followed in the article) but no further forms corresponding to 3, 4, 5, etc. Michael Hardy 00:45, 7 March 2006 (UTC)[reply]
  • 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)[reply]
  • 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)[reply]
  • Merge and Delete. Whatever new content here should be merged and this article deleted. -- Alpha269 14:39, 6 March 2006 (UTC)[reply]
    • 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)[reply]
  • Keep per MadCow257. --Siva1979Talk to me 14:52, 6 March 2006 (UTC)[reply]
  • 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
    • I've taught math a five universities, including MIT for three years; I think I have at least some ability to discern gibberish from valid content. I wrote this article. I believe any mathematician will understand it. It is absurd to speak of whether it is better or worse than the article titled mathematical induction in a way that presumes the two articles have the same purpose. This article does indeed contain information not in that other article. Michael Hardy 22:04, 6 March 2006 (UTC)[reply]
      • GWO is probably overstating his point, and I doubt he meant to attack your own ability, which is self-evident. The page history does not make clear who wrote it. (I maintain a delete vote on grounds of purpose, not quality.) Melchoir 22:37, 6 March 2006 (UTC)[reply]
  • Delete No material seems salvagable for merger. JoshuaZ 16:16, 6 March 2006 (UTC)[reply]
  • Delete- there is nothing here that isn't covered elsewhere. Reyk 21:11, 6 March 2006 (UTC)[reply]

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

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

*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)[reply]

Comments

This article was intended to be comprehensible to all mathematicians.

It was not intended to teach mathematical induction. It was not intended to explain what mathematical induction is, nor how to use it.

What I see is (mostly) a bunch of non-mathematicians looking at the stub form in which the article appeared when it was nominated from deletion, and seeing that

  • It is not comprehensible to ordinary non-mathematicians who know what mathematical induction is, and
  • The article titled mathematical induction is comprehensible to ordinary non-mathematicians, even those who know --- say --- secondary-school algebra, but have never heard of mathematical induction.

And so I have now expanded the article far beyond the stub stage, including

  • Substantial expansion and organization of the introductory section.
  • Two examples of part of the article that is probably hardest to understand to those who haven't seen these ideas.
  • An prefatory statement right at the top, saying that this article is NOT the appropriate place to try to learn what mathematical induction is or how to use it, with a link to the appropriate article for that. It explains that you need to know mathematical induction before you can read this article.

Therefore, I invite those who voted to delete before I did these recent de-stubbing edits, to reconsider their votes in light of the current form of the article. Michael Hardy 23:20, 6 March 2006 (UTC)[reply]

I don't think it's appropriate to post this argument to the talk page of every person who voted delete. Simply posting it here is enough. There is another issue that I don't think anyone has brought up -- what keeps this from being original research? I don't see any indication in the article that the idea of organizing different forms of induction this way is published and notable. --Allen 23:34, 6 March 2006 (UTC)[reply]
I agree with Allen on both counts. dbtfztalk 23:36, 6 March 2006 (UTC)[reply]
I am uncomfortable with the OR argument, since it doesn't seem to be strictly applied to most math articles. My beef is that the new material is still in the wrong place-- and this time, it doesn't even belong at Mathematical induction. A generalization of the product rule belongs at Product rule, and a generalization of the triangle inequality belongs at Triangle inequality. Mathematical induction should have a few sentences of prose concerning these strategies and a sentence summarizing each one, including a link. Melchoir 00:08, 7 March 2006 (UTC)[reply]
True, articles on basic mathematical concepts don't normally require references, but in this case, given all the hullabaloo, I think it's reasonable to request at least one. dbtfztalk 00:22, 7 March 2006 (UTC)[reply]

There is good reason to post on users' talk pages: many will not come back to look at this page after they've voted on it, so they won't see these comments. They can remove it from their talk pages; one user did that and cited, in his edit summary, the fact that it's a duplicate of what is here.
The three forms of course have been published separately, and the examples are well-known and routine. If anything original is here, it is the juxtaposition of forms of proof by mathematical induction, for the purpose of showing differences and similarities. Maybe this article is the closest I've come to posting something potentially publishable that may not have been published, but it would not be accepted as "original research" by journals devoted to publishing original research in mathematics. Only in a journal primarily devoted to expository articles as opposed to research article could an article on this topic have any hope of being accepted, unless it adds something hifalutin that is not hinted at here. Michael Hardy 00:25, 7 March 2006 (UTC)[reply]

As for the claims that the generalizations of the product rule and the triangle inequality belong in the articles on those topics: even if they should be there, the particular four-point form of analysis of the proofs belongs here, not there. That is most of the material on those examples. Michael Hardy 00:40, 7 March 2006 (UTC)[reply]

I agree that the four-point form does not belong in any other article, while I also believe that this article does not belong on Wikipedia. An insistence on that form without a reference leads me to start suspecting that this is all original research after all. The insight that (3 isn't the sum of 3 numbers less than 3) is relevant is honestly fascinating, but it too feels like original research. And getting back to my own motif, Polya's example is well-known enough to serve as an example of induction gone wrong in... you guessed it, Mathematical induction. It can even serve as a lead-in to the other two arguments, proof sketches of which should go in the relevant articles. The four-point form isn't necessary, since it boils down to saying "sometimes n = 2 gets us everything else"... and that phrase doesn't need an article of its own. Melchoir 01:07, 7 March 2006 (UTC)[reply]
Four-point form? It should also be pointed out that, in the triangle inequality example, the hard part is the step from 2 to 3, not the step from 1 to 2.
— Arthur Rubin | (talk) 21:56, 7 March 2006 (UTC)[reply]
Well, no. The "hard part" is proving the function is a metric in the first place, i.e., the case n = 2. That part is of course different in different metric spaces (and easy in some). Michael Hardy 23:02, 7 March 2006 (UTC)[reply]
No, again. That's n=3. n=1 is almost trivial () and n=2 is trivial. — Arthur Rubin | (talk) 23:15, 7 March 2006 (UTC)[reply]

Several people added their votes BELOW the "comments" section:

  • 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:
    • Of the three types of induction given, the first is the standard form and the third is the complete form, while the second is not a proof by induction at all, but either a technique of false proof (as in Polya's example) or a minor "start at b" variant of the first type combined with a baffling insistence on treating a trivialization of the theorem as an important separate case.
    • Let's look at the examples. The first two, the product rule and the triangle inequality, are isomorphic in the sense that the same thing is said in each one. Furthermore, they are isomorphic in the excesses they perpetrate: they discuss propositions for arbitrary integers which, when reduced to the case n = 2, give well-known theorems, but in the proof of the proposition, they begin by noting a trivial degeneration of the proposition. This is not a base case of the induction, precisely because the second case does not follow from the first, and furthermore, it gives entirely the wrong impression of these sorts of generalizations: in practice, a mathematician would prove the theorem by induction starting at the n = 2 case, and then note in passing that when n is artificially set equal to a case, such as 0, which was not covered, the statement remains vacuously true (this phenomenon is sometimes useful in discussions of generalities, or in the degeneration of other formulas).
    • The real problem with including these two examples in the context of Polya's is that while the first two examples illustrate real theorems, Polya's example illustrates a false theorem, yet the author presents the three examples in parallel as though there is some mathematical identification between them. I suppose the point he could be trying to make is that "if the transition from the first to the second case is impossible, you have to remember to explicitly check that the second case is true", but that's saying nothing more than that the second case is in fact the base case of the induction and that you have to be careful not to assume that the base case is, in fact, n = 1...which is just the "start at b" argument. Furthermore, he misrepresents the nature of the error in Polya's problem: it is not so much a question of whether the first-to-second transition is impossible (though of course that is ultimately what wrecks the proof) but whether Polya's glib characterization of the relationship between and is in fact correct. He also claims that this uses the n = 2 case, but that is not true: it follows from the assumption that the statement holds for n that it holds for 2; how you got to n is another matter, but actually, the general step of the induction makes no reference to the base case. The failure is ultimately in the case n = 2, but I feel like at least half the lesson that Polya was trying to teach was that reasonable statements can conceal unreasonable facts.

In summary, my feeling is that this page is entirely too pedantic in concept to appeal at all to a mathematician (since any mathematician understands induction quite well), but of no interest whatsoever to a layperson because it concerns an extremely ephemeral point of logic that is only useful in mathematical practice. Furthermore, the presentation is poorly executed and the material doesn't include even enough mathematical sophistication to make the point well. At best it's a usage guide, which is not encyclopedic. In parting, I'd also like to note (since this has been an issue in this discussion) that whatever I have to say about mathematical sophistication concerns the article, not the author. Ryan Reich 04:19, 7 March 2006 (UTC)[reply]

    • Well, I'm surprised at how many people that I thought would know better are missing the point. In practice, a mathematician might prove the generalization of the product rule or the triangle inequality by saying that it's a trivial induction on n. Of course the base case is n = 2; I thought I made that clear in the article. But this is not just a "start at b" where b = 2; rather it has a special form: 2 is the largest finite value of k such that k is not a sum of fewer than k numbers that are smaller than k, and that is why the proof takes the form it does. Polya's example does not illustrate a false theorem; Polya's example illustrates the general form. Of course the specifics he playfully puts into it make the proposition false. That serves pedagogical purposes when posed as a find-the-error exercise. But the main point of inclusion of Polya's example is that it clearly shows the form without the things that vary from one case to the next. Michael Hardy 21:16, 7 March 2006 (UTC)[reply]
  • 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)[reply]
  • merge. Form 2 can be easily be rendered as form 1, by a simple renumbering. Define two sequences of cases: let the original cases and let , be a renumbered sequence. So the form 2 induction for just becomes a form 1 induction for , with an extra vaciously true case . 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)[reply]
      • "Form 2 can be easily be rendered as form 1, by a simple renumbering." That is utterly false. The fact that a certain set has two members would then be relied on in a absolutely crucial way in the step that would have been renumbered so as to be called "1". A property of the number 2 would still be relied on at that step in an essential way. Michael Hardy 21:23, 7 March 2006 (UTC)[reply]
    • For the record, I am not in the practice of retaliating in any way after being reverted. This AfD is an honest attempt to remove an article from Wikipedia; at least during the nomination, "merge" and "delete" were essentially equivalent. Being attacked only made it more important that I seek out community consensus, and even if Michael had been civil, I've been around just long enough to know that a private bilateral debate with the owner of any article is a Bad Idea. In fact, the best way to obtain a civil discussion would have been to AfD this from the very beginning. Melchoir 17:11, 7 March 2006 (UTC)[reply]
  • 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 . (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)[reply]
  • 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)[reply]
  • 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)[reply]
    • 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)[reply]
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