I refactored some of the main deletion page here as it was growing too long to scroll through. Stifle 09:54, 9 March 2006 (UTC)[reply]

... and you've left numerous instances of erroneous math uncorrected on the project page, and accusations there, that are now responded to only here. Could you put a notice and a link to this discussion page after each of those? Michael Hardy 22:35, 9 March 2006 (UTC)[reply]

• • • [Melchoir's edits] 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]
      • I believe it is possible, reasonable, necessary, and consistent with the existing article to contrast forms of mathematical induction in Mathematical induction. The incompleteness of Mathematical induction is unfortunate, and Three forms of mathematical induction does not help.

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]

• • Here's an exercise for you:

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]

• • 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]

• • Have you ever heard the terms "how Archimedes used infinitesimals" or "how directness of sunlight causes warmer weather" or "list of 10 longest-reigning Popes" anywhere besides Wikipedia? If not, does that make them original research? Michael Hardy 21:04, 7 March 2006 (UTC)[reply]
  • Uh yes, people talk about how Archimedes used infinitesimals and even use that phrase; I even went to a talk a while ago that had that as one of the major topics. And yes to the second topic also. As for the third, no I generally do not hear things like that or even list of lists of mathematical topics. I fail to see the relevance. Perhaps you somehow got the impression that I construe any article with titles that are unusual as OR. That is not the case. Whether or not the term "three forms of mathematical induction" is common, however, is entirely relevant to whether the topic of that name is OR, as the topic is about "Proofs that a subset of { 1, 2, 3, ... } is in fact the whole set { 1, 2, 3, ... } by mathematical induction usually have one of the following three forms." --C S (Talk) 23:48, 7 March 2006 (UTC)[reply]

• • 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 ${\displaystyle \{1,\dots ,n\))$ and ${\displaystyle \{2,\dots ,n+1\))$ 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]
    • After reading your comments I went back and reread the article with them in mind. If I have done it correctly, your point is:in the "type 2" induction, the method of proceeding from case n to case n + 1 is to recast the case n + 1 statement as a combination of a case n statement with a case 2 statement, both of which are already proved.

    In the product rule case, this is done by splitting the product of n + 1 factors into a product with n factors and the additional factor, the two of which together constitute a product with two factors, and then apply the standard Leibniz rule. In the triangle inequality case, you take the sum of n + 1 distances and recast it as a sum of n distances with a remaining one, substitute for the former using the induction hypothesis, and then apply the standard triangle inequality. And in the Polya example, you split the set of horses in the same way.

    However, if this was your point it was totally lost on me in the article because you never made it explicitly. The article is heavily loaded with jargon that does nothing to improve the clarity of the presentation: for example, using "trivial" for the induction step is pretty egregious, in my opinion, as well as not illustrating the manner in which you find it trivial, which if I've read the article correctly is the entire point you were trying to make! You confuse the presentation with your insistence on actually explicitly mentioning the artificial "base" n = 1 case. The fact that the article manifestly assumes its point to be obvious makes its tone a little condescending in combination with the multiple, equally unelaborated, nearly identical examples. So at the very least I think the article is badly written. I also still think it's not encyclopedic:

    • Your choice of division into "types" of mathematical induction is still arbitrary. In fact, it seems even more arbitrary now that I understand what you are trying to say. Types 1 and 3 are the definition of induction in its incomplete and complete forms. Type 2 employs Type 1 for its logic and simply does so by a particular method. If you are so interested in the fact that 2 is the largest integer n not a sum of fewer than n lesser integers, then perhaps you would also be interested in other number-theoretic related proofs. For example, proof where the base case(s) are the prime numbers, which are not products of lesser integers.
    • The reason the number 2 is important here is that mathematics is filled with binary operations (like product, sum, and union, to name your examples). Why that occurs is like you say: because if you take any n-ary operation you can reduce it to a sequence of binary operations, and you can't reduce "binary" to "unary" since, well, 2 is not the sum of fewer than two lesser numbers. So part of the phenomenon is that in the Peano axioms, special preference is given to the pair of a number and its successor. Part of it is also that mathematics studies operations which are typically generalizable both up and down: we like things that can be done to two objects, and we like to be able to, yes, inductively define them for larger collections as well. If you look at this article in this way, your point is simply that for formulas whose dependence on n is in the length of iteration of a binary function, propositions tend to be provable by the methods of "type 2" induction. I am less than surprised.
    • On top of all that, I had to think way too hard to come up with this explanation. It smacks of original research, a sort of philosophical musing, but I doubt you could find any source that actually discusses this method of induction.
    • However, you could probably find a source that discusses the method of inductively defining functions: you know, we have f(n) defined for small n, and we use this to define f(n + 1). This discussion, on Wikipedia, is probably located (if present at all) in one of the foundations articles. It would be pretty relevant, there, to note that when such functions are involved, such a pattern of argument tends to occur in proofs by induction (it might still be original research; however, if it is used as an example rather than a statement of theory, it would be acceptable as well as more enlightening). Outside of even this context, however, this article becomes an isolated curiosity.

    Basically: I still think the article doesn't belong, but I think you could express its point effectively elsewhere. Ryan Reich 01:10, 8 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]
    • Merge and delete are not equivalent; they are polar opposites, both with regard to the history of the article and what happens to the present text. Septentrionalis 04:13, 8 March 2006 (UTC)[reply]
      • In my above comment, I am explaining my past actions. "Merge" and "delete" were essentially equivalent back when there was virtually nothing to merge. Salix alba lamented that AfD is overkill for merging, but this was never a classical merge. Melchoir 04:37, 8 March 2006 (UTC)[reply]

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. dbtfz^talk 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. dbtfz^talk 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.

${\displaystyle d(x_{1},x_{n})\leq \sum _{k=1}^{n-1}d(x_{k},x_{k+1})}$

— 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 (${\displaystyle d(x,x)=0}$) and n=2 is trivial. — Arthur Rubin | (talk) 23:15, 7 March 2006 (UTC)[reply]

No, it's not n = 3. The index n should be the number of terms on the right. As I keep saying, you can't just arbitraly reindex (except in the trivial and obvious sense). Michael Hardy 00:27, 9 March 2006 (UTC)[reply]

I don't see how you can avoid calling it n = 3 except by reindexing to make n = 0 a valid (and marginally interesting) statement. Similarly, getting generalized associativity from associativity, or "orderlessness" (at least the term in Mathematica) from the operator being commutative and associative (hmmm, both n = 2 and n = 3 are the real steps, there, while the induction step beyond that is trivial). (As an aside, I do recall a paper I co-wrote in which we started transfinite induction from -1, so there may be something to be said for this reindexing.) — Arthur Rubin | (talk) 00:50, 9 March 2006 (UTC)[reply]

I'm not sure which talk/deletion page this should go on, but I've written up a reformulation of Michael Hardy's arguments that, I believe, gets at the fundamental reason for his Second Form of induction. This reformulation (possibly abridged!) could be put in the mathematical induction article, letting us delete the Three Forms article. (The First and Third Forms are already treated well in the mathematical induction article.) I just came up with this, so it's not exactly well-considered; let the critique commence. Joshuardavis 01:49, 8 March 2006 (UTC)[reply]

In mathematics, many standard functions, including operations such as + and relations such as =, are binary, meaning that they take two arguments. Often these functions possess properties that implicitly extend them to more than two arguments. For example, once addition a + b is defined and is known to satisfy the associativity property (a + b) + c = a + (b + c), then the trinary addition a + b + c makes sense, either as (a + b) + c or as a + (b + c). Similarly, many axioms and theorems in mathematics are stated only for the binary versions of mathematical operations and relations, and implicitly extend to higher arity versions.

Suppose that we wish to prove a statement about an n-arity operation implicitly defined from a binary operation, using mathematical induction on n. Then it should come as no surprise that the n = 2 case carries special weight. Here are some examples.

In this example, the binary operation in question is multiplication (of functions). The usual product rule taught in calculus says

${\displaystyle (fg)'=f'g+g'f.\,}$

For a product of n functions, one has

${\displaystyle (f_{1}f_{2}f_{3}\cdots f_{n})'\,}$

${\displaystyle =(f_{1}'f_{2}f_{3}\cdots f_{n})+(f_{1}f_{2}'f_{3}\cdots f_{n})+(f_{1}f_{2}f_{3}'\cdots f_{n})+\cdots +(f_{1}f_{2}\cdots f_{n-1}f_{n}').}$

In each of the n terms, just one of the factors is a derivative; the others are not.

The n = 1 case is trivial (${\displaystyle f'=f'}$), and the n >= 3 cases are easy to prove from the preceding n − 1 cases. The real difficulty lies in the n = 2 case, which is why this is the one stated in the standard product rule.

In this example, the binary relation in question is equality, =, applied to color of horses.

In the middle of the 20th century, a commonplace colloquial locution to express the idea that something is unexpectedly different from the usual was "That's a horse of a different color!". George Pólya posed the following exercise: Find the error in the following argument, which purports to prove by mathematical induction that all horses are of the same color:

If there's only one horse, there's only one color.

Suppose within any set of n horses, there is only one color. Now look at any set of n + 1 horses. Number them: 1, 2, 3, ..., n, n + 1. Consider the sets {1, 2, 3, ..., n} and {2, 3, 4, ..., n + 1}. Each is a set of only n horses, therefore with each there is only one color. But the two sets overlap, so there must be only one color among all n + 1 horses.

Again the n = 1 case is trivial (any horse is the same color as itself), and the inductive step is correct in all cases n >= 3 cases. However, the logic of the inductive step is incorrect when n = 2, because the statement that "the two sets overlap" is false. Indeed, the n = 2 case is clearly the crux of the matter; if one could prove the n = 2 case, then all higher cases would follow from the transitive property of equality.

This example is somewhat more complicated, because it involves a binary operation, +, and a binary function, the distance function d in a metric space.

The usual triangle inequality in metric spaces says

${\displaystyle d(x_{1},x_{3})\leq d(x_{1},x_{2})+d(x_{2},x_{3}).\,}$

For a sequence of n points, one has

${\displaystyle d(x_{1},x_{n})\leq d(x_{1},x_{2})+\cdots +d(x_{n-1},x_{n}).\,}$

Again the n = 1 case is trivial, since

${\displaystyle d(x_{1},x_{1})\leq 0.}$

In this example the n = 2 case is also trivial:

${\displaystyle d(x_{1},x_{2})\leq d(x_{1},x_{2}).}$

Because two binary functions are involved, with the "middle argument" ${\displaystyle x_{2))$ matching, the first nontrivial case occurs when n = 3. This case is the crux of the matter, which is why the triangle inequality deals with it. The cases n >= 4 then follow inductively from the corresponding n − 1 cases.

Well, I think that in the triangle inequality one should let n be the number of terms in the sum on the right and that that should be recognized not to be a mere matter of convention. Yes you can trivially reindex, but what I'm saying is that the special property of the number 2 applies to the case indexed as n = 2 precisely when one does it that way. Michael Hardy 02:41, 8 March 2006 (UTC)[reply]

But it is a mere matter of convention. — Arthur Rubin | (talk) 02:43, 8 March 2006 (UTC)[reply]

I agree with Michael. It is, abstractly speaking, a matter of convention, but in the context of binary operators it is perfectly natural to let n be the number of terms in the sum. Also, this is the number of legs in the polygonal path whose vertices are the points appearing in the inequality, which makes it likewise natural to index on this quantity. Although I suppose you could argue that n should be the number of sides in the polygon. Still, it doesn't help the example to insist on an indexing which fails to bring out the essential feature it's trying to exemplify. Ryan Reich 02:58, 8 March 2006 (UTC)[reply]

Reindexing is fine; go ahead. This restores n = 2 to the crucial case, and makes the arity of the + operation the key consideration. What I'm really concerned about is whether the people interested in preserving this article's content would be satisfied by merging an explanation similar to this into mathematical induction. We can polish it later. Joshuardavis 03:33, 8 March 2006 (UTC)[reply]

"Mere convention": Only a few months ago I was surprised to find out that some respectable mathematicians think the fact that the empty product is 1 is merely convention. And that that topic is not worth discussing. Maybe it would not be worth discussing if it were mere convention. But it's beginning to look to me like a big gap in most mathematicians' understanding of what seems like trivial matters, and perhaps so is this. More shortly.... Michael Hardy 22:31, 8 March 2006 (UTC)[reply]

Is your point about conventions that they are chosen to preserve larger patterns in mathematics (and therefore, as a practical benefit, simplify notation)? Because I think that people who say "it's just a convention" know this. They know that the convention is not fully arbitrary, that in fact it's the only one that's "right". What they are really saying is that the conventional case (such as the empty product being 1) does not have the same obvious intuitive meaning as the typical case. Joshuardavis 02:58, 9 March 2006 (UTC)[reply]

All the discussion above has led me to be inclined to dispose of this whole matter in the following way:

• Redirect this to a separate article that would be primarily on the topic of the "second form", taking care to explain why one cannot "just reindex" in the relevant sense (trivially, one can reindex, but that's not the relevant sense).
• Emphasize on that other page that Polya's argument is not primarily about fallacious reasoning, but rather, it exemplifies the form of this argument. I'll probably cite a book by Polya, and one by a Russian author whose name is escaping me at this moment.
• Maybe I will emphasize the reasons why this form is not trivially reducible to the others.

I'm surprised that a substantial amount of confusion has appeared among mathematicians who at first said you can just reindex (showing a firm grasp of the obvious while missing the point), or that some of this is merely convention. I put a footnote in a recent paper that said:

Perhaps as a result of studying set theory, I was surprised when I learned that some respectable combinatorialists consider such things as [the fact that the number of partitions of the empty set is 1] to be mere convention. One of them even said a case could be made for setting the number of partitions to 0 when n = 0. By stark contrast, Gian-Carlo Rota wrote in \cite{Rota2}, p. 15, that "the kind of mathematical reasoning that physicists find unbearably pedantic" leads not only to the conclusion that the elementary symmetric function in no variables is 1, but straight from there to the theory of the Euler characteristic, so that "such reasoning does pay off." The only other really sexy example I know is from applied statistics: the non-central chi-square distribution with zero degrees of freedom, unlike its "central" counterpart, is non-trivial.

So maybe this kind of point is not as widely understood among mathematicians as I thought. I'll be back Thursday or Friday..... Michael Hardy 00:22, 9 March 2006 (UTC)[reply]

Can you come up with some precise formulation of a criterion that distinguishes proofs by this kind of induction from those that aren't? (And even better, can you source it in the literature?) I still have a sense that you're making a sort of Scholastic distinction among various arguments, a distinction that you've never really made precise. As such, as I said, it's a good point to pass along to your students, but I'm not convinced it's encyclopedic, and it also has an OR-ish flavor.

(That's not to say that Scholastic distinctions can never be encyclopedic, but I'm certainly more likely to want sources for them than for precisely formulated mathematics.) --Trovatore 02:31, 9 March 2006 (UTC)[reply]