Comments on Is ‘How would you know to do the next step?’ always a bad question?
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Is ‘How would you know to do the next step?’ always a bad question?
We have a user who keeps posting questions of the form, ‘How would you [tortured synonym for “know”] to [do the next step in a proof]?’ Leaving aside the various other reasons that these posts are bad[1], my question is whether the question is intrinsically bad based on its form alone. (If so, I would assume that the policy should be to close any such question, as no amount of editing to solve the other problems would save it.)
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I don't want those other issues to be a distraction from the central question here. I'm only mentioning them so that people can consider what these posts might look like if they were otherwise written to be exemplary questions: if they showed understanding of or engagement with the surrounding context (well-researched), if they rewrote the central concepts in their own words instead of (or perhaps in addition to) posting large screenshots of the source material, if they were written using a vocabulary that wasn't needlessly obtuse and distracting (good English), and if they were questions about mathematical concepts instead of questions about understanding the non-mathematical parts of what an author is communicating (basic reading comprehension). ↩︎
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Absolutely not. They're the kind of questions as https://math.stackexchange.com/questions/tagged/motivation?tab=Votes and https://math.stackexchange.com/questions/tagged/intuition?tab=Votes.
So the most time mathematicians are working, they're concerned with much more than proofs, they're concerned with ideas, understanding why this is true, what leads where, possible links. You play around in your mind with a whole host of ill-defined things.
I can't remember which book. If you know, edit my post. Field Medallist Michael Atiyah wrote somewhere.
Is there one big question that has always guided you?
I always want to try to understand why things work. I’m not interested in getting a formula without knowing what it means. I always try to dig behind the scenes, so if I have a formula, I understand why it’s there. And understanding is a very difficult notion.
People think mathematics begins when you write down a theorem followed by a proof. That’s not the beginning, that’s the end. For me the creative place in mathematics comes before you start to put things down on paper, before you try to write a formula. You picture various things, you turn them over in your mind. You’re trying to create, just as a musician is trying to create music, or a poet. There are no rules laid down. You have to do it your own way. But at the end, just as a composer has to put it down on paper, you have to write things down. But the most important stage is understanding. A proof by itself doesn’t give you understanding. You can have a long proof and no idea at the end of why it works. But to understand why it works, you have to have a kind of gut reaction to the thing. You’ve got to feel it.
And somewhere else, Atiyah wrote ...
So the most time mathematicians are working, they're concerned with much more than proofs, they're concerned with ideas, understanding why this is true, what leads where, possible links. You play around in your mind with a whole host of ill-defined things.
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