Timeline for Thinking and Explaining
Current License: CC BY-SA 4.0
19 events
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| Jan 5 at 2:13 | history | edited | Michael Hardy | CC BY-SA 4.0 |
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| Jan 4 at 15:49 | comment | added | David White | I want to point out to any new users who stumble upon this answer that the method on display here, of a back-and-forth conversation between the OP and the answerer, via discussion in comments and editing of the answer, is very unusual on MO and generally frowned upon, because it keeps bumping the thread. We often say "MO is not a discussion forum." This thread was an exception because of the people involved. | |
| Jan 10, 2020 at 7:18 | history | edited | Hans Lundmark | CC BY-SA 4.0 |
Fixed broken link to “Mumford's treasure map”.
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| Oct 26, 2017 at 9:38 | comment | added | Jose Brox | @MinhyongKim In your final addition, you are seeing the integers "just" as a subdirect product of the $\mathbb{Z}_n$. Subdirect products happen to be very useful in ring theory for looking at elements as sequences of elements from better rings (when having a nontrivial infinite subdirect product involved). For example, any semiprime ring is a subdirect product of prime ones (I realize you surely know all this better than me...) | |
| Oct 20, 2015 at 16:24 | history | edited | Todd Trimble | CC BY-SA 3.0 |
fixed broken link
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| Sep 16, 2010 at 15:59 | comment | added | Minhyong Kim | Agreed! Thank you very much for your comments. Please give my best regards to Reyer Sjamaar and Ravi Ramakrishna. | |
| Sep 16, 2010 at 3:22 | comment | added | Bill Thurston | After final addition: I think you should mention your way of thinking to students and colleagues. It may seem like a small thing, but small things are important, and I suspect it would help set the right frame of mind. Isn't this method used in computer algorithm for computations with large integers? That could help motivate/justify it, not that there aren't more mathematical justifications. I'm enjoying this conversation, which obviously could continue for a long time following threads that have started, but this being MO, we should probably interrupt it and wait for other chances. | |
| Sep 16, 2010 at 2:37 | history | edited | Minhyong Kim | CC BY-SA 2.5 |
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| Sep 16, 2010 at 0:18 | history | edited | Minhyong Kim | CC BY-SA 2.5 |
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| Sep 15, 2010 at 15:50 | history | edited | Minhyong Kim | CC BY-SA 2.5 |
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| Sep 15, 2010 at 15:20 | history | edited | Minhyong Kim | CC BY-SA 2.5 |
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| Sep 15, 2010 at 14:00 | comment | added | Bill Thurston | After "Added again": When listening to a lecture, I can't possibly attend to every word: so many words blank out my thoughts. My attention repeatedly dives inward to my own thoughts and my own mental models, asking 'what are they really saying?' or 'where is this going?'. I try to shortcut through my own understanding, then emerge to see if I'm still with the lecture. It's the only way for me, and it often works. For something like Spec(\mathbb Z) (for me, now), I need extra guidance to avoid diving into unhelpful places. Perhaps explicit instructions "don't think __1, think __2", and why. | |
| Sep 15, 2010 at 13:47 | comment | added | Cam McLeman | If only I could give a +1 to each of the three thirds of this post. The "money in the bank" analogy is particularly fantastic. | |
| Sep 15, 2010 at 13:02 | history | edited | Minhyong Kim | CC BY-SA 2.5 |
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| Sep 15, 2010 at 4:38 | comment | added | Bill Thurston | The example you added to your answer is very telling. As an outsdier to the field, I have only a vague sense of the mathematical meaning of Mumford's picture, but I think I understand its role and its effect that you described. It's important to ask yourself "Why?" when others don't comprehend something of this nature. Usually it's because they have competing mental models where your words are nonsense. It requires some excavation of the old before you can embrace the new. | |
| Sep 14, 2010 at 23:42 | history | edited | Minhyong Kim | CC BY-SA 2.5 |
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| Sep 14, 2010 at 22:39 | comment | added | Minhyong Kim | Perhaps it wasn't clear from what I wrote that I have quite strong urges to wax philosophical, and maybe many people do. I probably also indulge in it more often than I realize. It's just that I understand perfectly when other people's eyes glaze over. Anyways, I was trying to describe a pretty broad, fundamental, and possibly reasonable disjunction between our sense of how things work and what we, with the implicit support of the community, allow ourselves to express. | |
| Sep 14, 2010 at 19:15 | comment | added | Bill Thurston | @Minhyong Kim: Thanks for your thoughtful and appropriate response to the question. I've been aware that when I drift too long into philosophizing or meta-discussions, people's eyes glaze over. My eyes often glaze over at that kind of stuff, as well. I don't advocate that people spend a lot of time philosophizing. I do advocate that we try to make their mathematical discussions informed by awareness of how they are thinking, and an attempt to see how it might appear to the other person. I think it can help us get to the point more quickly and effectively. | |
| Sep 14, 2010 at 15:22 | history | answered | Minhyong Kim | CC BY-SA 2.5 |