[qarl2]

Hmmm... seems to me that if you can find a solution without creating the desired explanation - then that's a problem with the original question - not the solution itself.

And discovering a bad question leads to the correct question. No?

[Garlef]

> then that's a problem with the original question - not the solution itself

I think there's a good counterexample to this:

Atiyah/MacDonald proove the Nullstellensatz ultimately by using some trick involving determinants.

They give a very nice theoretical treatment of the content and context of the theorem. But the proof at one crucial point uses techniques that live conceptually outside of this context: While its possible to see that the argument is sound, it does not give a good explanation of _why_ it's true within the context of the theorem.

(You could of course argue that they did not give enough context ... but that's exactly my point: the trick makes the proof work but hides the explanation)

[wbl]

I was baffled enough by this comment to take my copy and look. The Nullstellensatz is an exercise late in the book long after Noetherian rings are introduced and they don't even do the Rabinowitz trick in the hints as they have enough theory to hit it the hard way. Determinants are nowhere to be found.

[Garlef]

It's been a few years (~ 15) since I read it.

The determinant trick I'm referring to is used in prop 2.4 (in the Version I found via Google).

They use the determinant of the adjugate matrix if I remember correctly... But they just hit the reader with this without any motivation or even naming it.

[wbl]

For Nakayama lemma (but they give a cleaner proof later). Note that the book was from the 1960's and from lectures at Oxford: matrix manipulations like that were a lot more commonplace.

[coldtea]

>(You could of course argue that they did not give enough context ... but that's exactly my point: the trick makes the proof work but hides the explanation)

Can't one see it in another way: that the trick illuminates a deeper explanation, connection the theorem's context and the stuff that's conceptually outside of that context. And that the problem is we don't know why the two domains (the context and the conceptually outside of it one) are related and cooperating in this way.

[qarl2]

Ah... is this just the difference between a constructive and non-constructive proof? Is that the distinction you're making?