tag:blogger.com,1999:blog-3722233.post4959127204319308129..comments2024-09-15T21:39:59.938-05:00Comments on Computational Complexity: When did Math Get So Hard- Part 2Lance Fortnowhttp://www.blogger.com/profile/06752030912874378610noreply@blogger.comBlogger9125tag:blogger.com,1999:blog-3722233.post-1308376516617637402023-11-03T07:02:59.001-05:002023-11-03T07:02:59.001-05:00The preprint has been scrapped after publication o...The preprint has been scrapped after publication on September 1, it is now behind a paywall.<br /><a href="https://philpapers.org/rec/WESIIO-2" rel="nofollow">Idealist Implications of Contemporary Science.</a><br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-78241363517444602602023-10-27T02:25:16.640-05:002023-10-27T02:25:16.640-05:00Ha! I sort of guessed that. I can't tell what&...Ha! I sort of guessed that. I can't tell what's "more advanced" in this rarified air universe, but I did notice that the main three prereqs were 1967, 1967, and 1977, which is an age ago in math terms.<br /><br />More generally though (for us beginners), math builds on things. One can't do Lie Algebras without linear algebra and modern algebra; you just can't. It very much sounds like all of modern math is like that. Classical number theory may have been different: my impression is that Ramanujan's work was more about brilliance in ability to manipulate stuff than in depth of structure. Maybe.<br /><br />But the bottom line is that modern math is about building structures on (and between (e.g. Langlands)) structures that came before. And one isn't going to play in the fast lanes without doing the work. For we amateurs, there's a lot of beauty in even the simple stuff, so there's pleasure to be had, but if you need it for work in another field, e.g. Comp. Sci., life is going to be harder...<br />David in Tokyonoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-79202295096990685662023-10-26T08:54:13.254-05:002023-10-26T08:54:13.254-05:00I assume an early year PhD studying number theory ...I assume an early year PhD studying number theory and arithmetic geometry is suppose to know these materials.The list does not include more advanced stuff such as etale cohomology.Alexander Grothendiecknoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-4795718210983013222023-10-26T08:09:03.544-05:002023-10-26T08:09:03.544-05:00The landscape of math (and math theorems) is like ...The landscape of math (and math theorems) is like a Mandelbrot set ... you can keep zooming-in and you'll always find new nice places. Marzio De Biasihttps://www.nearly42.orgnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-18430767817605080902023-10-25T21:48:48.345-05:002023-10-25T21:48:48.345-05:00I feel that the distinction between a good mathema...I feel that the distinction between a good mathematician and a random graduate student is that mathematicians somehow manage to read and comprehend all of the prerequisites precisely. It is possible to have some ideas without a solid foundation, but it is impossible to make a real contribution.Ryan Ilponoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-48124454242518251772023-10-25T07:44:46.936-05:002023-10-25T07:44:46.936-05:00In graduate school on the first day of one class, ...In graduate school on the first day of one class, the professor listed the prerequisites, then noted that no one had all of these prerequisites. One of the things that you learn how to do in graduate school is to learn stuff when you don't have all the prerequisites. Basically, you fill in the ones you need as you need them. Of course, it helps to already have some of the prerequisites.David Marcushttps://www.blogger.com/profile/07084520656051241766noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-69163925755581075132023-10-24T10:53:22.727-05:002023-10-24T10:53:22.727-05:00"While we might hope that math should be more...<i>"While we might hope that math should be more tractable because its an abstraction that we humans created, the reality points in the opposite direction."</i><br /><br />Actually there are good reasons to believe that the so-called "reality" is <i>also</i> a creation of the human mind:<br /><br /><a href="https://www.academia.edu/download/105582108/westerhoff_idealist_implications.pdf" rel="nofollow">Idealist Implications of Contemporary Science.</a> (PDF)<br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-28666610260005687912023-10-24T10:25:48.721-05:002023-10-24T10:25:48.721-05:00Real things are really hard, super-duper hard; in ...Real things are really hard, super-duper hard; in fact its supposed to be exactly that way! <br />Imagine creating living things including humans from bare-bones organic compounds on a primordial earth.<br />It's not like chemistry back then was any different from what we get to work with today.<br />It took nature ~4 billion years of constant toil to get us from there to here; to figure out how to reliably retain what works and relentlessly discard the stuff that does not work over this whole period.<br />While we might hope that math should be more tractable because its an abstraction that we humans created, the reality points in the opposite direction. space2001https://en.wikipedia.org/wiki/2001:_A_Space_Odysseynoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-64515559919713847572023-10-24T06:02:29.580-05:002023-10-24T06:02:29.580-05:00Math is hard because ALL of the "structure&qu...Math is hard because ALL of the "structure" is to be funneled through human mind (as your post well exemplifies) and there is no <i>full formalization</i> which could use processing power to tame the volume.<br />And don't tell me about proof assistants, they are ridiculously complex and limited and add to the problem instead of alleviating it.<br />Anonymousnoreply@blogger.com