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amichail
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proof by magic
« on: Nov 1st, 2005, 4:07am » |
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I think that one problem with proofs in research papers is that they are succinct, with little remaining of the thought process that led to those proofs.
And so it seems that the proof just appeared by magic, with little intuition as to how one might come up with such proofs.
One might expect textbooks and survey papers to provide some idea of a line of reasoning that could lead to a proof.
But how would they do this?
Do most people think in similar ways? Or perhaps the idea is to use textbooks and survey papers as an opportunity to get people to think in similar ways -- a common vocabulary of thought patterns?
Would there typically be only a few lines of thought that would lead to a proof in practice (i.e., even if there are many possible avenues to a proof, perhaps only few tend to arise with most people) -- and thus it is those few lines of thought that should be described in textbooks & survey papers? How would you find them?
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| « Last Edit: Nov 1st, 2005, 4:34am by amichail » |
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towr
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Re: proof by magic
« Reply #1 on: Nov 1st, 2005, 8:09am » |
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Making proofs is general part insight, part strategy, part luck, and a few quarts of sweat.
Luck, you either have or you don't.
Insight, comes from practise.
Strategies is really the only thing you can try to examine. And the first thing of course is to try and capture the proces of proving. Make a few proofs, but don't just make them, look at "how could you do this in general".
What sort of proofs are we talking about anyway? Logic, maths, correctness of programs?
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amichail
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Re: proof by magic
« Reply #2 on: Nov 1st, 2005, 2:34pm » |
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on Nov 1st, 2005, 8:09am, towr wrote:Making proofs is general part insight, part strategy, part luck, and a few quarts of sweat.
Luck, you either have or you don't.
Insight, comes from practise.
Strategies is really the only thing you can try to examine. And the first thing of course is to try and capture the proces of proving. Make a few proofs, but don't just make them, look at "how could you do this in general".
What sort of proofs are we talking about anyway? Logic, maths, correctness of programs? |
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Let's say we consider algorithm design, perhaps trying to find tight lower and upper bounds for a problem.
I guess the point is this: is it worthwhile to try to reuse thought patterns, even ones that are quite vague, ad hoc, and complicated? Or is it sufficient to just study the final cleaned up succinct proof?
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| « Last Edit: Nov 1st, 2005, 2:35pm by amichail » |
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towr
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Re: proof by magic
« Reply #3 on: Nov 1st, 2005, 3:03pm » |
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on Nov 1st, 2005, 2:34pm, amichail wrote:| I guess the point is this: is it worthwhile to try to reuse thought patterns, even ones that are quite vague, ad hoc, and complicated? |
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If you want to make new proofs, then yes.
Quote:| Or is it sufficient to just study the final cleaned up succinct proof? |
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If you just want to demonstrate correctness of a result, then yes.
I'd say a cleaned up proof is preferable if you only want to show that your result is right. As long as people can get from each step to the next.
You don't really want to burden them with all idiosynchrasies in your original proof. And if you make a detour that in hindsight isn't necessary and just makes for a longer proof, it's better left out.
On the other hand, if you want to show how you can arive at a proof, how to construct one. Then I'd go with the strategies that led you there. They may be ad hoc, or only be heuristics, but they're worthwhile tools. Of course, then you should also show how you can clean up the proof afterwards.
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amichail
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Re: proof by magic
« Reply #4 on: Nov 1st, 2005, 3:09pm » |
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on Nov 1st, 2005, 3:03pm, towr wrote:
If you want to make new proofs, then yes.
If you just want to demonstrate correctness of a result, then yes.
I'd say a cleaned up proof is preferable if you only want to show that your result is right. As long as people can get from each step to the next.
You don't really want to burden them with all idiosynchrasies in your original proof. And if you make a detour that in hindsight isn't necessary and just makes for a longer proof, it's better left out.
On the other hand, if you want to show how you can arive at a proof, how to construct one. Then I'd go with the strategies that led you there. They may be ad hoc, or only be heuristics, but they're worthwhile tools. Of course, then you should also show how you can clean up the proof afterwards. |
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Suppose you are writing a textbook. Which strategies do you focus on? Perhaps there are many ways to arrive at a proof. Shouldn't your choice of strategies to present depend on how typical students think? And if so, how do you determine how typical students think?
Also, what about unsuccessful strategies? Do you present those? And if so, how do you pick the ones to present?
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Icarus
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Re: proof by magic
« Reply #5 on: Nov 1st, 2005, 9:11pm » |
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One example of the phenomenon you refer to: For centuries, mathematicians have wondered about Archimedes' proof that the volume of a sphere is (4/3)[pi]r3. It is evident from the proof that Archimedes knew the value before he started, and used it in guiding the proof. So the question was, how did he figure it out in the first place?
Then sometime in the 20th century (I've forgotten exactly when, but I am sure it was recent history), a forgotten manuscript was discovered which described a lost manuscript by Archimedes called "The Method".
In it, Archimedes notes that the sum of the cross-sectional areas of a hemisphere and a right-circular cone, both of radius r, with the hemisphere section taken at a distance x from the planar end and the cone taken at the same distance x from the vertex, will add up to the area of a circle of radius r (the sphere cross-section has area [pi](r2 - x2), while the cone cross-section has area [pi]x2).
So he considered comparing the cone and hemisphere to a cylinder of radius and height r. Since the cross-section at x has the same area for the cylinder as for the cone & hemisphere combo, regardless of the value of x, Archimedes reasoned that the volumes of the hemisphere and cone together must add up the the volume of the cylinder. Hence the volume of the hemisphere had to be ([pi]r3 - ([pi]/3)r3) = (2/3)[pi]r3).
Archimedes rejected this as an approach for an actual proof, though, apparently because he had a hard time accepting the idea of volume being constructed of infinitesimal slices of area. (As well he should. We only accept it today because more conventionally founded mathematics leads to the conclusion. For Archimedes to have based a proof on the idea without that earlier development would have been unacceptable.) Therefore he created the proof he gave out, which estimates volume using myriads of conic pieces.
Had his Method been given with the proof, it might have led other mathematicians to examine why it worked, and therefore led to the development of calculus much earlier than Newton and Liebnitz.
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towr
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Re: proof by magic
« Reply #6 on: Nov 2nd, 2005, 1:22am » |
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on Nov 1st, 2005, 3:09pm, amichail wrote:| Suppose you are writing a textbook. Which strategies do you focus on? |
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I'd have to know/examine which there are first.
But they have to be understandable of course. Not too complicated (depending on the level the textbook is for, or seperated in basic and more advanced chapters).
Quote:| Perhaps there are many ways to arrive at a proof. Shouldn't your choice of strategies to present depend on how typical students think? |
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Sure, and especially how you want them to think after reading the textbook.
How they think will influence how you can teach them. And how much of their thinking needs to be changed to understand how to tackle a problem.
Quote:| And if so, how do you determine how typical students think? |
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Cognitive research. f.i. get them to try and solve problems, while thinking out loud. (Which is one way to discover problem solving strategies that are used).
Or make tests that can highlight which path someone takes to a solution. But then you have to map out a lot of possible strategies beforehand, and think up tests to uncover them (f.i. by making problems that can easily be solved by the one strategy, but not by the other)
Quote:| Also, what about unsuccessful strategies? Do you present those? And if so, how do you pick the ones to present? |
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If you discover a strategy that is often employed, but also generally fruitless, I think it'd be a good idea to present it.
"This is what people often do, and it generally goes wrong here, for this and that reason"
People often have to learn what not to do as much as what they should do.
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rmsgrey
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Re: proof by magic
« Reply #7 on: Nov 2nd, 2005, 10:00am » |
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on Nov 2nd, 2005, 1:22am, towr wrote:If you discover a strategy that is often employed, but also generally fruitless, I think it'd be a good idea to present it.
"This is what people often do, and it generally goes wrong here, for this and that reason"
People often have to learn what not to do as much as what they should do. |
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A couple of counter-arguments for airing common misconceptions:
Putting misconceptions into someone's head could just confuse the issue for them. The risk is that they remember the misconception and forget that it's wrong.
There are so many potential misconceptions that any attempt to list them all (or even the most common ones) will leave the wrong approaches outnumbering the valid ones.
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towr
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Re: proof by magic
« Reply #8 on: Nov 2nd, 2005, 1:34pm » |
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It's true that there's a problem there. But on the other hand, if someone already has the wrong idea, you need to be able to explain to him why it's wrong. I've often found teachers can't do that without help (in fact on several occasions I found out what I did wrong, and why it was wrong, before them, and left them behind slightly confused). So perhaps it should be in the teachers textbook rather than the students textbook.
I suppose it depends on whoever uses the textbook, and how well it's written. And what domain, naturally.
If you're dealing with propositional logic you can't neglect to explain that A or B, isn't an exclusive or. And that "affirmation of the consequent" is not a valid logical proof rule. That's so common a misconception it needs to be adressed.
(I suppose you could argue neither is actually a strategy, but they do allow you to "proof" certain things fairly quickly if you use them.
I'm sure better examples exist.  )
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Grimbal
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Re: proof by magic
« Reply #9 on: Nov 2nd, 2005, 3:26pm » |
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So it is a good thing to give exercises to the students. They will try plenty of wrong approaches just by trying to find a proof. Then the teacher can show the "right" approach.
Well, that means that in the head of the students, it feels like "what I do is usually wrong, what the teacher does is usually right". Not very motivating, but at least, it helps to keep the two well separated in the student's minds.
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rmsgrey
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Re: proof by magic
« Reply #10 on: Nov 4th, 2005, 4:44pm » |
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on Nov 2nd, 2005, 3:26pm, Grimbal wrote:So it is a good thing to give exercises to the students. They will try plenty of wrong approaches just by trying to find a proof. Then the teacher can show the "right" approach.
Well, that means that in the head of the students, it feels like "what I do is usually wrong, what the teacher does is usually right". Not very motivating, but at least, it helps to keep the two well separated in the student's minds. |
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It's better to let the student explain to you why they're wrong - when you have the time. Presenting situations where the student's misconcieved version conflicts with their established knowledge hopefully prompts them to reevaluate.
If nothing else, getting students to grapple with a problem before exposing them to the "magic" solution gives them a framework into which to fit the new concepts, making it more likely they'll understand and remember them.
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Barukh
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Re: proof by magic
« Reply #11 on: Nov 6th, 2005, 12:30am » |
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This is a very interesting topic. I am strongly convinced that while explaining the path to the solution, “wrong paths” should be also mentioned. This turns an otherwise dry explanation to an exciting narrative, and also gives the reader an expression that nobody is perfect.
One of the most important principles of this forum is: “Explaining how you arrived at an answer is more valuable than the answer itself.” And this no doubt includes approaches that did not lead to the answer.
G. Polya wrote wonderful books on the subject: “How to Solve It” (brief “summary” may be found here) and “Mathematics and Plausible Reasoning, Volume 2: Patterns of Plausible Inference”.
There is also a related question about aesthetics of the solution. Once a solution is found, does it matter how beautiful it is? How much material is needed for it? Etc… J.E. Littlewood in his book “A Mathematician's Miscellany" attributes to A. S. Besicovitch the following statement: “A mathematician's reputation rests on the number of bad proofs he has given.” It is meant that the very first proofs are often cumbersome.
P. Erdos called God the “Supreme Fascist” (SF). He said “the SF has this transfinite Book … that contains the best proofs of all mathematical theorems, proofs that are elegant and perfect." The strongest compliment Erdos gave to a colleague's work was to say, "It's straight from the Book."
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| « Last Edit: Nov 6th, 2005, 2:21am by Barukh » |
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