Timeline for Generalizing a problem to make it easier
Current License: CC BY-SA 2.5
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| Nov 18, 2010 at 16:33 | comment | added | roy smith | I suspect there is some confusion here between the relative ease of proving a theorem in a more general setting and actually thinking up the idea. I have always found it easier to think of a solution in a more special case, but then once the problem is understood, it is easier to separate out the crucial parts, and give them in a general setting. Mumford told me even Grothendieck worked this way. He would begin from a simple idea, and reflect on it until he had placed it in its most general possible setting. There is also the dichotomy between conjecturing a solution and proving it. | |
| Nov 2, 2010 at 15:39 | history | edited | roy smith | CC BY-SA 2.5 |
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| Oct 31, 2010 at 21:14 | comment | added | roy smith | I agree it is possible to find an overly special solution to a special case, but in my experience it does not happen that often and the opposite happens more. A little example is the Torelli theorem for curves of genus 4 by intersecting the tangent quadric and the osculating cubic at the unique pair of conjugate double points of the theta divisor. Then the idea of intersecting all the tangent quadrics at all double points in higher genera is not too great a stretch. I agree that there are people who more easily grapple with more general versions of a problem but I am not one of them. | |
| Sep 28, 2010 at 18:59 | comment | added | Nick S | True, but what if when one makes the problem more special, the extra information is competelly irrelevant for the problem, and more it is also missleading. Very simple example (probably not the best): Let $a,b,c >0$. Prove that $\sin(a) + \sin(b)+ \sin(c) \leq \frac{a^3+b^3+c^3}{abc} \,.$ This problem has two obvious trivial generalisations, and in both of them it becomes pretty clear that it is irrelevant that the $a,b,c$ on left/rigth sides of the inequalities are the same, but many students could be misslead by this fact on the wrong path. | |
| Sep 28, 2010 at 4:29 | history | answered | roy smith | CC BY-SA 2.5 |