[HN Gopher] Mathematicians Transcend Geometric Theory of Motion ___________________________________________________________________ Mathematicians Transcend Geometric Theory of Motion Author : theafh Score : 93 points Date : 2021-12-09 15:28 UTC (7 hours ago) (HTM) web link (www.quantamagazine.org) (TXT) w3m dump (www.quantamagazine.org) | echopurity wrote: | It's unfortunate that they omit Floer's suicide. Nobody wants to | talk about the negative effects of a career in mathematics. | sabellito wrote: | You're saying that an article about "Mathematicians Transcend | Geometric Theory of Motion" should also be about "the negative | effects of a career in mathematics"? | kevinventullo wrote: | Are mathematicians statistically more likely to commit suicide? | pizza wrote: | Genius and dead at 34 - if you read between the lines, it said | it all. | moelf wrote: | >Arnold predicted that every phase space of a certain type | contains a minimum number of configurations in which the system | it describes returns to where it started. | | is this article talking about Ergodicity without mentioning | Ergodicity? https://en.wikipedia.org/wiki/Ergodicity | ABeeSea wrote: | All dynamical systems have a phase space and ones with the | return property don't necessarily have to be ergodic. | [deleted] | revskill wrote: | What if mathematical theorems are described by executable code | instead of just symbols then ? Currently it seems impossible for | reader to verify all theorems in a math paper. | joppy wrote: | This question seems to always come up on Hacker News - figuring | out how to even formulate the statement of theorems in theorem | provers such as Lean is a massive research undertaking in | itself, requiring much creativity and novel work. Let alone | figuring out how to formulate the proofs. I'd recommend you | have a look at some of the talks that Buzzard has given on this | to understand the complications (both technical and social) and | progress that has been made so far. | eigenket wrote: | Currently transforming a new piece of maths from its "standard" | form (i.e. what working mathematicians actually use) to | something a computer can understand is a big task that usually | takes a team of experts something of the order of years. | | Maths is really hard and proofs require a tonne of steps. For | this reason mathematicians have to be comfortable jumping over | the standard pedestrian intermediate steps in proofs and just | focusing on the important stuff. This is necessary because | including all the details would obscure the important stuff | (imagine directions for driving somewhere with steps like "now | walk up to the car", "now click the opener", "now open the car | door", "now sit down in the drivers seat"). | | Computers (currently) are way too dumb to skip these steps so | you have to walk them through it. | r-zip wrote: | There are efforts in that direction: | https://leanprover.github.io/theorem_proving_in_lean4/ | ABeeSea wrote: | A major advancement happened this year where a recent paper | from a fields medalist on an incredibly abstract topic was | formally proved in Lean. The theorem was that Scholze's new | condensed mathematics was logically consistent with real | functional analysis. | | https://www.quantamagazine.org/lean-computer-program-confirm... | del_operator wrote: | Well, this is how I learned Manolescu left UCLA and is now at | Stanford | del_operator wrote: | Also, yet another article that makes me rethink my choice to | take Homological Algebra with no real algebraic topology | coursework. Merkurjev was teaching so at least I have all his | notes. It kind of forced me to give up any concrete basis and | just handle abstract tools. | ABeeSea wrote: | If you already understand homological algebra, Peter May's AT | book might get you where you want to be quickly. It's free on | his website. | mathematicaster wrote: | good advice | spekcular wrote: | I urge you to read Hatcher's book (or better, tom Dieck's | book published by EMS) immediately. I can't even imagine | learning homological algebra without a bunch of concrete | topological examples to compute with. That sounds confusing. | xyzzyz wrote: | Homological algebra without algebraic topology or geometry | sounds like the driest possible exercise in pointlessness. | wrycoder wrote: | Tell them what you're going to tell them. | | Tell them. | | Tell them what you told them. | | The first and last parts are missing from this article. You have | to take a gestalt approach and scan the whole thing to get an | idea of what it's about. | munificent wrote: | I despise this structure and the kind of writing and | presentations it tends to produce. The absolute worst, which I | see all too often in presentations, is its fractal form: | 1. Tell them that you are going to: 1. Tell them | what you are going to tell them. 2. Tell them. | 3. Tell you what you told them. 1. Tell them what you | are going to tell them. 2. Tell them that now you will | tell them what you will tell them. 3. Tell them. | 4. Tell them that you are done telling them what you will tell | them. 5. Tell them that you will tell them what you | told them. 6. Tell them what you told them. 7. | Tell them that you are done telling them what you told them. | | A structure I like much better which permits the above | structure but allows other variations is: | | 1. Explain how to tell if this is worth their time. | | 2. Tell them. | | 3. If there was a lot, suggest what's worth remembering. | | The focus here is on _how it benefits the audience_ and not on | some arbitrary structural form. | blablabla123 wrote: | It's also a trade-off between length of presentation or | article in that case and amount of interesting content. Also | I think Dynamical Systems are quite a mixture of theory and | practical applications, so it makes sense to mix that. | enobrev wrote: | It seems you're suggesting the same strategy, and just don't | like when it's done poorly. | munificent wrote: | Not at all. There are many ways to let an audience decide | whether the rest of the material is worth their time. | Summarizing to give them a preview is only one (and often | the least interesting one). | | Other ways: | | * Describe a problem that the audience also has, so that | they understand that you are aligned with them. | | * Tell an engaging anecdote so that they expect it will be | a rewarding experience. (The idea that a piece of writing | should entertain, inform, or persuade and that those are | mutually exclusive is another canard that I find to be | completely toxic and antithetical to good writing. Good | writing should entertain, inform, _and_ persuade.) | | * Telegraph that the time investment will be smaller by | getting started and making the overall thing shorter. | | * Describe previous failures to solve a problem. | | * Give them an interesting insight right off the bat, which | implies there may be more to come. | | * Tell them something personal which conveys whether you | are likely to be a person with interesting things to say. | | * Throw out a detailed, hard to acquire fact, which implies | that you have other hard-won knowledge. | | Note that what all of these have in common is that the | intro material _is unique_ and is not simply a pre- | statement of information they will encounter lately. | | Also, the fact that I made step 3 optional is significant. | Most writing and presentations don't need a summary and a | summary will often detract. If you want to stick in the | audience's memory, what you really need is a _climax_ , and | "here's what I just said, said again" is about the most | anti-climactic ending you can imagine. | [deleted] | crispyambulance wrote: | It's an article and not a power-point slide. There's nothing | wrong with the format. | | The outline you gave is more appropriate for slide-deck talks | where the audience is captive and they're apt to be unconscious | during the middle. | adrianmonk wrote: | The reason I'd like part 1 (tell them what you're going to | tell them) is it helps me answer the question, "Is this | relevant and interesting enough to me to spend the time | reading it?" | | In a sense, I need part 1 _because_ I 'm not a captive | audience. If this were (say) a lecture in a college class, | then it's a foregone conclusion that I'm using the time, so I | might as well pay attention. | | But since it's a web article, I have the choice to keep | reading or close the browser tab. I'd prefer to be able to | make an informed choice. | canjobear wrote: | It's also the standard for scientific papers. | wisty wrote: | I think science has the abstract (tell em what you'll tell | em) then the body and a discussion which is more "OK, now | that I got you're attention, here is what I actually think | but can't prove". | crispyambulance wrote: | Right, but it's also not a scientific paper. | eigenket wrote: | That depends on the field, especially in areas of pure | maths I see this "slide-deck" style way less. | hprotagonist wrote: | there isn't a universal standard for scientific papers. | | more's the pity. | Ar-Curunir wrote: | Science is a very broad term, and writing style varies | between research groups, let alone between fields. | rbanffy wrote: | Sometimes the journey is its own destination. | ABeeSea wrote: | I like the way quanta writes their math articles and I hope | they never go towards a stilted formulaic approach to writing. | :shrug: | smitty1e wrote: | > The planet's position and momentum can be described by six | numbers, three for each property. If you represent each of the | different configurations of the planet's position and momentum as | a point with six coordinates, you'll create the phase space of | the system. In this case, it has the shape of flat six- | dimensional space. The motion of a single planet can be | represented as a line weaving through this space. | | Sounds like ephemeris? | | https://en.m.wikipedia.org/wiki/Ephemeris | paulpauper wrote: | I think the gulf between research math and teaching math is so | wide that it may as well be a different subject altogether. The | vast majority of mathematicians are teachers, not researchers. If | all mathematicians have at the very minimum PHDs, then is the | difference in ability so great? | spekcular wrote: | It's not so hard to get a PhD in math, in the sense that if | you're willing to attend a low-ranked program and have at least | a moderate affinity for mathematics, you could probably do it. | It takes a lot of time to learn all the prerequisites, but | that's why the undergrad degree is 4 years and the PhD | typically 5+. Then you just find a suitable advisor, ask to be | handed a dissertation problem and some ideas for the solution, | write down that solution, and graduate. Anyone who's been doing | math research for a few years has a collection of problems they | know how to solve but haven't written up for various reasons, | which they can give for this purpose. (Usually: the question is | too boring or simple, no one cares, and there are bigger impact | things to do instead). | | Doing research that meaningfully advances mathematics, as | opposed to being make-work in service of getting a degree? Much | harder. | vecter wrote: | I wouldn't consider someone a mathematician unless they were a | researcher. Otherwise what does a mathematician do? Many people | have undergraduate or advanced degrees in math, but most of | them don't "do math" for a living (i.e. research). | [deleted] | physicsguy wrote: | Same for most science - physics, chemistry, etc... I have a | Physics PhD and I still wouldn't consider myself a physicist | as applying the physics skills is not what I do day to day. | bckr wrote: | with a bachelor's in Biochem, in my head I say I'm formally | trained in biochemistry. But iut loud I just say "my | bachelor's was in biochem" | mathematicaster wrote: | often as large as a draft prospect and nba player with years of | experience | syki wrote: | I was ABD in math at a top 25 program when I went to a talk | given by a graduate student from Berkeley. He was in my area | and we had the same length of them spent on the subject. As | close to equals in terms of experience and area of study as one | can get. He was far better than me and I knew that I'd never | understand the subject as well as him. Shortly after I quit the | Ph.D. program. I realized I would never do anything worthwhile | in the field. There is a large variation in talent within the | community of professional mathematicians. Outliers amongst | outliers. | | In trained in MMA for a number of years and sparred against | some local fighters. They were better than me but I could get | some hits in. I could cause them to expend some effort. Once I | sparred with a low level UFC fighter. He thoroughly destroyed | me. It was like I was 5 years old. Outliers amongst outliers. ___________________________________________________________________ (page generated 2021-12-09 23:00 UTC)