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https://mathoverflow.net/questions/71691/geometric-meaning-of-a-trigonometric-identity

In the question on mathoverflow that is linked above, I mentioned that I had proved an identity by mathematical induction. Noam Elkies, a professor at Harvard, posted an answer. He did not answer the question actually posed. Rather, he posted a better proof of the identity that I had proved by induction.

Is his proof actually better? My inclination is to say "yes", but I'm not staking my life on that.

But some years ago, I concluded that when a proposition can be proved either by mathematical induction or by other methods, the proof by other methods is usually better. This was based in part on various particular examples. But I can't remember what any of those are!

So was I right? And if so, what are (1) the examples (hundreds of them, if you have them!), and (2) the explication of how they are better?

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    What does "better" / "inferior" mean? I'm highly tempted to close as "not a real question".2011-09-03
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    Would the person who voted to close please explain? @Zev: This looks like a reasonable question to me. Similar questions on [proofs by contradiction](http://math.stackexchange.com/questions/240/) have created a lively and highly interesting discussion, I don't see an a priori reason why this should be different here. For what it's worth, **I vote against closing.**2011-09-03
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    On the other hand, I would prefer to convert this to a CW since the last part of a question invites a big list.2011-09-03
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    @Zev: A reasonable interpretation of ‘inferior’ might be ‘requires a stronger logical system, i.e. more axioms’; we might also interpret ‘better proof’ as meaning ‘proves a stronger result’. I think this question could have interesting answers.2011-09-03
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    I don't think it should be closed either, but maybe it would work well as community wiki due to its subjective nature. (For what it's worth, my first inclination is to interpret "better" as indicating that more insight is provided by the proof.)2011-09-03
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    @Zhen: I agree that that interpretation would be an interesting question, but that was not the sense that I felt that Michael meant - I read it as specifically using the subjective, vague sense of "better". Perhaps I'm mistaken; at any rate some clarification from Michael would greatly help this question, I think.2011-09-03
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    @Theo: Basically to reiterate my comment above, in the example you give AgCl specifically mentioned that he was thinking about logical strength. Many of the (excellent) answers also focused on why proofs that are not by contradiction usually give more insight; that's also a great measure of a proof. But simply "better" is not, and I was only (admittedly, over)reacting to the lack of detail about what is meant by "better" here.2011-09-03
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    @Zev: I can't read Michael's mind... I don't think that it needs to be restricted to a purely logical interpretation. In combinatorics there is the notion of a [bijective proof](http://en.wikipedia.org/wiki/Bijective_proof) and those are often preferred to other proofs since they lead to more insight to the problem at hand. Look at the beginning of Stanley's [Enumerative Combinatorics](http://www-math.mit.edu/~rstan/ec/ec1.pdf) where a considerable amount of effort is invested in explaining this philosophy (starting on page 19). I think there can be a variety of interesting answers.2011-09-03
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    The server serving @Theo's above link to *Enumerative Combinatorics* is intermittently returning 404 Not Found errors -- the link is correct; it will eventually provide the file if you keep trying. (However in the current version the discussion of bijective proofs seems to start on page 21.)2011-09-03
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    @Theo: I absolutely agree - as I say in my comment, logical strength and insight are both essential measures of how "good" a proof is (in fact, I think one could argue they are the only ones). And an interpretation of the question that I would love to see answers to would be whether there is a philosophy about induction proofs similar to the philosophy of bijective proofs in combinatorics. I would simply prefer that the question actually *say* something along these lines. I overreacted initially, I agree that closure is not appropriate here, but2011-09-03
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    what I don't like in the question is use of "better" as if it were obvious what it meant, and the only issue is determining whether it applies to induction proofs or not.2011-09-03
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    A proof is a proof and certainly not better or worse than any other per se if you consider only the "prove sth" aspect. What you maybe think of is a typical proof pattern (at least for identities): "Guess and prove by induction." This is not a weak proof, but it is fair to say (imho) that it is an unfair one. It does not tell you how to solve a similar problem; the solution might have been wildly guessed, or engineered, or devised in a rigorous way, you don't know. In fact, I know of one professor who used to *only* publish such proofs in order to keep a monopoly on his methods.2011-09-03
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    One should note that induction has numerous applications outside of this pattern.2011-09-03
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    There's a second vote to close; I thus join Theo and Jonas to vote against closure.2011-09-03
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    I, like Theo, Jonas and J. M., vote against closure.2011-09-03

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