[HN Gopher] Math writing is dull when it neglects the human dime...
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Math writing is dull when it neglects the human dimension
Author : mathgenius
Score : 191 points
Date : 2024-03-29 02:03 UTC (20 hours ago)
(HTM) web link (golem.ph.utexas.edu)
(TXT) w3m dump (golem.ph.utexas.edu)
| bedobi wrote:
| I'm no mathematician, so I only took basic school math, but I
| hated every moment of it. Mostly because overwhelmingly there was
| never any context or justification for learning any of it. Why
| does this exist? What actual real world problems does it solve?
| How did folks come up with and it, prove it works and start using
| it? Crickets. Just learn this formula, then that. The first time
| I heard the ancients calculated the distance to and size of the
| moon with trigonometry I was floored. Oh ok so that's the kind of
| cool shit they came up with it for. Now I'm listening.
| loloquwowndueo wrote:
| Sounds like you had crappy teachers.
| latency-guy2 wrote:
| Establish why its "crappy teacher" and not "crappy student".
| I've seen far more of the latter than the former.
|
| I don't think its hard to find something interesting about
| math either, and it is immediately applicable, even with what
| I think is extremely stupid in the form of common core as
| presented in the USA, you are literally being presented story
| problems about common every day occurrences and activities.
| atoav wrote:
| As someone who routinely teaches (Non-STEM) university
| students practical applications of math I can assure you
| that none of the students that told me "I am bad at maths"
| went out of my class without an understanding of the topics
| we talked about.
|
| Yet I routinely hear a: "Wow, if they told it to me this
| way during school I might have cared!" or "It really made
| my head smoke, but it felt good."
|
| Step 1 is believing that every person that doesn't have
| cognitive problems can understand an abstract concept if it
| is explained well and they got the motivation to understand
| it. Then you need to create a situation which motivates
| them to understand it and now you only need to explain it
| well.
|
| Many math teachers fail already at step 1. They believe 90%
| of their students are too stupid to understand things,
| while this is just a convinient explaination for their own
| failures. I once was convinced of being that student.
|
| I met _one_ kid that I couldn 't teach anything because he
| would forget things I told him 15 minutes befoee. He was a
| refugee from Afghanistan and severely traumatized.
|
| The truth is that we should look to the best when teaching
| and we would be stupid if we didn't. And the best are
| people like 3blue1brown. If your class falls significantly
| below such a level of clarity and engagement it will suck.
| During my own math education I had teachers that left out
| the most fundamental applications. E.g. something like an
| integral has clear applications, it is a new super power
| with which you can solve new problems -- yet all we did was
| learning a receipe and solving abstract problems. The fact
| that it was a super power was something I had to figure out
| myself, later.
|
| And this was common. I even was lucky because I had a good
| physics teacher who managed to being up a lot of what we
| learned in math and give it a more practical feel, but most
| of my friends from other schools were not so lucky.
| bedobi wrote:
| This is exactly how my education went. Oh you're not
| engaged with my shitty teaching and forcing you to do
| endless abstract nonsensical formulae with zero
| contextualization or application? Then you're the problem
| - you have cognitive defects, you are plain dumb and
| stupid, you are disinterested, you don't care about your
| education etc etc so I cannot help you. Now let me spend
| my time on "teaching" the kids who mostly already know
| what I'm "teaching".
| pylua wrote:
| That is a consistent problem with the education system no
| matter what the topic is with few exceptions.
| mycologos wrote:
| I think the economic returns to "decent understanding of X
| plus decent communication skills" are much higher when X is
| math than when X is art or language or history, so you need a
| greater passion for teaching to select it in spite of that
| fact, and this shrinks the pool of good math teachers
| relative to other subjects.
| bobajeff wrote:
| I'm pretty sure math is the most poorly taught subject out of
| all of them. Social studies, current events, history,
| literature arts are mostly skills most people use everyday.
|
| Math is important so every school is required to teach it
| however not many schools can. The issue with math is there
| are never going to be very many good math teachers as that
| would require many more people who know math. How many adults
| even know math beyond basic arithmetic?
| melagonster wrote:
| Even the basic content from textbook is not real
| mathematics. the daily work of mathematician is very
| different to calculate some interesting things.
| melondonkey wrote:
| Hard to meet everyone where they are and at the same time give
| them a relevant practical application for their own life. Good
| learners just soak it up and look for the application later.
| But that also doesn't fit all. It's hard to even write a pop
| song that everyone likes so math education that appeals to all
| is almost impossible
| skhunted wrote:
| I've been teaching mathematics for decades. Know how many
| students like the cool applications at the time they are taking
| the class? Very, very few. Usually, it's later on in one's
| journey through life that appreciation for the cool
| applications occur. At the time of taking the class doing cool
| applications inspires 1, pisses off 50 because it's too hard,
| and leaves 49 rolling their yes.
| atoav wrote:
| Haven't thought is for decades, but was my environments
| favourite math tutor, because I managed to ground everything
| in people's reality and rell them a compelling fiction in
| which knowing how to do the thing was actually a super-power.
|
| You know like the survival tricks certain preppers learn and
| never use -- that, but with math. Even if the examples were
| sometimes over the top, involved flaming moats and other
| xkcd-like freakiness I got students where their teachers told
| me it is hopeless sitting there participating with glowing
| eyes.
|
| Meanwhile in my own math education I had a teacher who made
| us do integrals for what felt like a year without telling us
| _once_ what the hell it is needed for. I had to figure that
| out on my own.
|
| My colleagues who didn't care just learned it by hard and
| forgot it immediately after. But I guess they got thought
| _all the material_ and got ok grades so their education was a
| success.
|
| What bullshit. Now, 15 years later I relearn a lot of those
| thinga because school made it sound boring only for me to
| later discover it is one of the most exiting things your
| brain can do. But that is about thinking on solutions to
| actual things not learning steps and doing them and getting
| punished for making one mistep.
| skhunted wrote:
| _What bullshit. Now, 15 years later I relearn a lot of
| those thinga because school made it sound boring only for
| me to later discover it is one of the most exiting things
| your brain can do. But that is about thinking on solutions
| to actual things not learning steps and doing them and
| getting punished for making one mistep._
|
| This is precisely what I was referring to about people who
| come back at a later time and relearn the stuff. They want
| applications. At the time a class is taken very few
| actually want to dive into applications. The subject has
| been taught the way it is taught for a reason.
| cultofmetatron wrote:
| That 1 will go and actually do amazing things with it. maybe
| invent a new application of the mathematics or invent new
| mathematics. The other 49 will forget it as soon as its no
| longer something to learn on an exam.
| fossuser wrote:
| imo this is largely because of the incentives around the game
| of school and testing.
|
| If you must wait 20yrs before you can do interesting stuff
| and you're evaluated primarily on your ability to maximize
| your grade then anything that gets in the way of that is an
| annoying distraction for most at best.
|
| Even if you're a student predisposed to find applications and
| the narrative of the discovery interesting you still have to
| focus on guessing what the test questions will be and just
| doing those, spending mental cycles on other stuff is a
| "waste" in that environment.
|
| Once you're finally free of school only then can you actually
| learn based on where your curiosity takes you, though at that
| point most choose not to - the curiosity having been driven
| out of them.
|
| Makes me think a little bit about the movie the lives of
| others:
|
| > "Did you know that there are just five types of artists
| Your guy, Dreyman, is a Type 4, a "hysterical
| anthropocentrist." Can't bear being alone, always talking,
| needing friends. That type should never be brought to trial.
| They thrive on that. Temporary detention is the best way to
| deal with them. Complete isolation and no set release date.
| No human contact the whole time, not even with the guards.
| Good treatment, no harassment, no abuse, no scandals, nothing
| they could write about later. After 10 months, we release.
| Suddenly, that guy won't cause us any more trouble.
|
| "Know what the best part is? Most type 4s we've processed in
| this way never write anything again. Or paint anything, or
| whatever artists do. And that without any use of force. Just
| like that. Kind of like a present."
| skhunted wrote:
| For the vast majority of the students the curiosity, as you
| put it, isn't really there. Understanding is very hard work
| and most people don't want to put in the work to acquire
| understanding.
| fossuser wrote:
| Perhaps, but it isn't helped that spending effort to
| understand is often in direct conflict with the work
| required to get a good grade.
|
| You're not tested on understanding how stuff is derived
| or how it's used, you're tested on grinding problems,
| particularly ones that are easy for teachers to put on a
| test and easy to grade (if they even bother with that, my
| worst teachers didn't create their own tests or grade
| them, machines did both). In English or humanities you're
| tested on predicting whatever bullshit your teacher
| believes and then crafting an essay that leans into their
| cognitive bias.
|
| I got very good at school and it was mostly by trying to
| model the minds of my mostly bad teachers and getting
| good at predicting what they want to hear and what they'd
| ask on the exam. With that down I could focus personal
| time on the stuff I was truly interested in (which
| ironically is what actually had market value).
|
| At least in the US public school system this is further
| hurt by public school teachers often barely knowing the
| material themselves and that's if they're not also
| outwardly hostile/condescending to the kids (there are
| always great teachers, but they're the exception to the
| rule).
|
| The system isn't selecting for the right things and those
| with enough money know this and work around it.
| tarkin2 wrote:
| If you look at the comments in khan academy there are tonnes
| of people asking for real life applications--that said,
| they're given in most of the material there.
|
| Inspiring students (and showing the material can help them in
| their lives additionally) is one of the most important but
| difficult parts of teaching: good explanations falling on
| uninspired ears rarely settle.
|
| And I'd argue the onus is on the teacher to inspire but it's
| not something I can say is an easy skill to master.
|
| Succinct explanations can be hammered out but fostering
| inspiration is a soft skill rarely taught---and rarely deemed
| important in the already inspired.
| skhunted wrote:
| I imagine - but have no data to back me up on this - that a
| lot of those comments you mentioned on Khan Academy come
| from two types of people. Those who are relearning the
| subject and are amenable to applications and those who say
| they want the cool problems but when they are actually done
| fall into the 99 category I mentioned above.
|
| It sounds to me like you haven't taught much. I could be
| wrong.
| Ekaros wrote:
| At university level not insignificant amount of students are
| there because they have to. The course is mandatory. And they
| might only want to pass. They are not getting degree because
| they want education, but because it is perceived as needed in
| society. Now should these be ignored or not is a discussion
| to have.
| bawolff wrote:
| > How did folks come up with and it, prove it works and start
| using it
|
| I mean, anything at the university level should include proofs
| on why it works. I would go as far as to say you aren't really
| doing math if there are no proofs.
| lupire wrote:
| _elementary_ level too! Humans crave understanding. This is
| what we finally have with modern materials like Eureka.
| golol wrote:
| Did you not have a physics class around the same time you
| learned calculus and linear algebra? That makes it obvious what
| the application is.
| constantcrying wrote:
| This is completely asinine.
|
| What you say applies to _every_ subject in school. I
| interpreted poetry, learned ancient history and dead language.
| Yet somehow the single most useful tool of thought humans have
| developed needs to justify itself so that you will learn it?
| eternityforest wrote:
| I did a (not fully third party reviewed for errors, watch out)
| project to record all the math related "cool shit I'm glad to
| have discovered":
| https://github.com/EternityForest/AnyoneCanDoIt/blob/master/...
| jrm4 wrote:
| It's really this. I was a "good" student so I got pretty far in
| college math; and _none_ of it has stuck with me without a real
| life application. At ALL.
|
| What's kind of killing me now, as my kids go through algebra et
| al, is that now we have a VERY OBVIOUS way to make this
| interesting and we're severely underutilizing it, which is
| video games. "Draw a rainbow in Minecraft" or "Figure out the
| trajectory of that frag grenade" seems just gobsmackingly
| obvious as a path here.
| Brian_K_White wrote:
| I always liked Lockhart's Lament
|
| https://maa.org/sites/default/files/pdf/devlin/LockhartsLame...
|
| It makes a very different point about teaching, or
| learning/discovering math, not writing about math.
| kouru225 wrote:
| Paul Lockhart was my high school teacher! He completely changed
| my mind about math.
| mycologos wrote:
| As somebody who works in a mathy subarea of computer science, oh
| man, I agree. My heart always falls when I need a result and it
| turns out the original paper is some terse typewritten notice
| from the 70s whose first sentence is a definition with a bunch of
| proper nouns and whose main theorem is given at the most general
| possible level with no applications at all.
|
| I have talked with math people about why this is, and responses
| are some combination of
|
| a) being concise and being elegant are the same, same for maximum
| generality/abstraction
|
| b) the people who should read the paper don't need things
| explained
|
| c) I am afraid that some smart egotistical professor whose
| opinion I value for some reason will call me soft if I add extra
| handholding material
|
| (Nobody has ever really said c, but my sense is it's true.
| Academic writing has a lot of imitation of style to prove you're
| part of the in-group.)
| saithound wrote:
| In my experience, (c) is a very large part of it. At one point
| in my career, I decided to try writing good, accessible
| articles, which properly motivated definitions and well-
| explained arguments with plenty of hand-holding.
|
| When I did that, a version of the derogatory sentence "The
| proofs are easy / non-technical." would appear in the reviews.
| Every. Single. Time. Of course, I have some independent
| confirmation that the proofs weren't easier than in any of my
| other work (e.g. my coauthors and I had to work just as hard to
| get them), but this led to having to resubmit them to less
| prestigious journals than the ones which normally published my
| work.
|
| I gave up on this approach, and realize now that the opposite
| is more likely to be rewarded: out of 18 eventually-published
| papers, I only managed to piss off the referees enough for a
| revise/resubmit decision once, and I really went out of my way
| to keep the proofs vague that time.
|
| Of course, I had a largely unremarkable career in a somewhat
| niche subfield: I'm sure there are levels where (a), (b), and
| more importantly the sheer speed required to get a result out
| are bigger incentives. And from yet other fields, I
| occasionally hear rumors of people who master the art of opaque
| writing and "parallel construction" only to make it difficult
| for others to get ahead of them (hi shashe!).
| abdullahkhalids wrote:
| My advice to all academics in STEM is, just write the main
| body of the paper exactly as how the orthodoxy demands. Use
| the style needed to get the paper accepted. Then, add a
| supplementary or appendix of the paper that is written for
| the human graduate student. Put in worked out examples,
| further details on the proof. Most times, it will just get
| through, and you will have accomplished your goal.
|
| If the journal demands you remove the appendix or
| supplementary, just remove it from the published manuscript.
| Then add it to your ArXiv submission.
| saithound wrote:
| This is good advice, but for a very different problem.
|
| 1. The problem in math is not that the way the "orthodoxy"
| insists on presenting things in a suboptimal way, but that
| if the reviewers find a good explanation of your result,
| they'll recommend that you publish in a lower-ranked venue
| than your result would ordinarily merit (at least unless
| you solved a famous problem). So researchers are
| incentivized to make the presentations of the proofs as
| opaque as possible. You can see this in conference and
| workshop talks (which tend to happen pre-publication in
| math), in many fields speakers always avoid presenting
| _any_ proofs. Putting better explanations in an appendix,
| which the referees can read, simply wouldn't help.
|
| 2. Being easy-to-understand at first is actively punished,
| but even being easy-to-understand in the long run is not
| rewarded. You can always write an explanatory blog post
| after the publication decision has been made, but you won't
| put effort into writing one if you don't gain anything from
| it. This applies even more to the people in the example,
| who wrote on a typewriter in the 1970s. There were no
| blogs, and it was much harder to get an appendix through
| because page limits were physical limits, and the act of
| writing was much more onerous before the age of computers
| and LaTeX. There was no point to doing it given that it was
| actively discouraged.
| abdullahkhalids wrote:
| I think this is fair criticism. I am coming for
| theoretical physics, where, at least in my area, I see my
| proposed strategy actually being practiced quite often.
| Physicists are quite a lot less grouchy than
| mathematicians, and there is a lot less of your point 1
| in Physics.
|
| That said, to present in "high impact" journals, you do
| have to write your results in some grand fashion where it
| is the greatest thing since sliced bread. But the
| results, not the proofs. Then again, the difficult
| theorem proving papers are rarely published in high
| impact journals.
|
| Also, I have seen quite a number of papers where the
| arxiv submission is more updated/expanded than the
| published version. If you have some new framework that
| you want people to adopt, then it is in your benefit that
| grad students actually understand the ins and outs of it,
| so people so inclined do put in some effort to make their
| results accessible.
| light_hue_1 wrote:
| > if the reviewers find a good explanation of your
| result, they'll recommend that you publish in a lower-
| ranked venue than your result would ordinarily merit (at
| least unless you solved a famous problem).
|
| Even if you solve a well known problem. I gave a talk
| once where I did so as a very junior student. During the
| Q&A a very well known senior person got up and basically
| asked me what's the point and this is all trivial anyway.
| Thankfully I remembered the exact place where the founder
| of the entire field had said just recently this is one of
| the hardest problems in the space and the problem he had
| hoped to maybe one day get to when he started the entire
| enterprise. The audience laughed and the guy apologized.
|
| But I learned my lesson. Talks and papers need a little
| magic. For some people you can't just solve a cool
| problem, they need to think that they couldn't have done
| so and that they don't quite get how you did it. I now
| include something in every talk that I don't expect the
| audience to get just so what I'm doing seems "hard" to
| people who think this way.
| GTP wrote:
| Thanks for sharing your experience, but it leaves a
| bitter taste in my mouth: although well motivated, I find
| the outcome to be unfortunate. In an ideal world, people
| would appreciate the elegance of a simple solution to a
| seemingly hard problem. But, as you pointed out, the sad
| reality is that to some people, if your solution doesn't
| look hard, it reflects negatively on the importance of
| your result
| solveit wrote:
| One problem is that many problems seem easier than they
| are, and you only find out that they're hard by failing
| to solve them. I have often wondered how many
| unremarkable foundational results would be considered
| major accomplishments if they didn't have the
| mis(?)fortune of being found by the first person who
| tried.
| Dalewyn wrote:
| I've always found math is taught in the most daft, bland,
| vapid, worthless ways imaginable and I've thought it had
| to do with those who do the teaching: The teachers and
| textbooks.
|
| But reading this comment chain, am I correct to
| understand that this problem stems from the very essence
| of math itself? The people who live and breathe math just
| _fucking hate_ sharing their passion with others?
|
| What the hell.
| gosub100 wrote:
| Could be the hazing mentality: I suffered to get here,
| why should I allow others to get here without suffering?
| lupire wrote:
| Academics (not just mathematicians) are famously bitter
| and political, because they do things tht the world
| values so little, but they care about so much, and they
| are fighting for scraps of recognition and funding.
| protomolecule wrote:
| Yeah, when the result is easy to understand people think it
| was easy to arrive at. What surprises me that smart people
| don't correct for that bias.
| GTP wrote:
| Just my opinion, but I think that actually smart people
| appreciate the elegance of a simple solution to a problem
| that looks hard at first sight. It's the people that aren't
| so smart but want to sound smarter that are incentivized to
| make their results look harder than what they actually are.
| didntcheck wrote:
| I guess it's analogous to the phenomenon of some people
| feeling "ripped off" when they pay a tradesman (or other
| worker) to do something that (appears) physically easy. As
| the apocryphal story goes "you're not paying me to turn a
| screw, you're paying me to know which one to turn" or "I
| could call my apprentice and have him take longer to do it
| if you'd like"
| keybored wrote:
| This is the HN IQ bias: assuming that smart people are
| somehow less susceptible to cognitive biases etc. that the
| mere mortals have.[1]
|
| Smart people can be incredibly biased and ideological. Some
| careers for smart people are even all about reasoning
| backwards from a given conclusion.
|
| [1] redacted footnote
| mayd wrote:
| > ... some terse typewritten notice from the 70s
|
| Personally, I rather like these these; they have a certain
| retro-appeal, in particular old Springer mathematics
| publications. We are so spoilt with LaTex.
| xelxebar wrote:
| While gatekeeping is definitely a thing, I really suspect it's
| not the major incentive here.
|
| In writing (both prose and code!) there is always a question of
| target audience, which inevitably excludes the not-target
| audience. Personally, for a field in which I'm an expert, it's
| really annoying to continually wade through introductory
| material and hand-holding just to get to some small nugget of
| substance. In that case, I'm definitely not the target
| audience, so I'll go looking for another communications channel
| that offers the compressed/elegant/general/abstract/terse
| formulations I desire. Please don't then insist that I'm being
| unfair and exclusionary if you're not the target audience on
| those specific communications channels.
|
| For math papers and whatnot, whitepapers are like the one
| established channel for experts, while everyone else has
| textbooks, introductory pamphlets, blogs, youtube videos, etc.
| I agree, however, that there are cases where non-experts could
| benefit from knowledge siloed within expert communication
| channels, but this is an unfortunate systematic side-effect not
| malice.
|
| Honestly, with software development, I find it disappointing
| that our social conventions currently conflate "readability"
| with "comfort and familiarity to Generic Programmer" instead of
| something more useful like "facilitates domain understanding
| and insight to the primary developers".
| mycologos wrote:
| The target audience point is fair. There are basic concepts
| that I don't bother defining in any paper I write, and I'm
| sure there are people who would make the same argument about
| them that I'm making here. But I'm not suggesting writing
| everything like "An Extremely Gentle And Slow Introduction to
| X, With Lots of Reassurances That You Can Do It".
|
| The question probably comes down to how accessible a paper
| should be. Personally, I think a reasonable bar is something
| like: a third-year PhD student in your broad area should be
| able to skim the paper and say a couple of paragraphs about
| what's happening and why it matters, and upon reading the
| paper more closely, present it in a seminar and defend it at
| least a couple of questions deep. IMO, most papers are not at
| this level, and are instead pitched at actual experts.
|
| I think my point (and TFA's point as well) is that going from
| an experts-only paper to a seminar-ready one is actually not
| a ton of text. It might increase a paper's length by 5%. It's
| not about making everything longer, but adding enough context
| and signposting that the story can be followed at multiple
| levels, from the 5-minute pitch you'd get at a poster session
| to the every-detail one the author has. So I think we can
| have a paper that both the grad students and experts like.
|
| I think doing this takes some skill and effort, which is well
| within the reach of most of the people who can write these
| papers, but _way_ less effort is expended on this non-
| technical aspect.
| kd5bjo wrote:
| I've started to keep a collection of readable papers
| (mostly engineering/experimental science, but a little bit
| of more theoretical work as well) that cover what I
| consider to be pretty foundational concepts, because
| additional citations are ~free and it at least gives an
| entry point for readers that are on the edge of the target
| audience.
|
| As a consequence, I've read papers that were written
| anywhere from the late 19th century to just a year or two
| ago. In my experience, the older papers tend to be more
| understandable at a conceptual level but more modern ones
| tend to be more precise with the details. There is likely
| some survivorship bias here, though, as there has been more
| time for the worst of the old papers to be forgotten.
|
| The writing style has also changed a lot-- The older papers
| present things in a much more narrative way, and it can be
| a challenge to bring the kind of motivational context that
| allows into a paper that would feel at home in a modern
| publication.
| Folcon wrote:
| > I've started to keep a collection of readable papers
|
| Are these shared / mentioned anywhere?
|
| A resource like this would be super helpful.
| markusde wrote:
| I would also be interested in reading some of these
| examples!
| nix0n wrote:
| > Honestly, with software development, I find it
| disappointing that our social conventions currently conflate
| "readability" with "comfort and familiarity to Generic
| Programmer" instead of something more useful like
| "facilitates domain understanding and insight to the primary
| developers".
|
| It's sort of a similar effect: most software is written by
| software people for other software people, just like most
| math is written by math people for other math people.
|
| I personally have had the experience more than once of a
| mathematician apologizing for their code style but then
| handing me code that I find to be more straightforward and
| readable than usual.
|
| What specific conventions do you think would facilitate
| domain understanding and insight?
| noelwelsh wrote:
| I think it's the influence of Bourbaki[1] who were the original
| mathematical edgelords. I have a book (Creating Symmetry[2])
| that explicits rejects this style. It's a lovely book.
|
| [1]: https://en.wikipedia.org/wiki/Nicolas_Bourbaki
|
| [2]:
| https://press.princeton.edu/books/hardcover/9780691161730/cr...
| constantcrying wrote:
| >My heart always falls when I need a result and it turns out
| the original paper is some terse typewritten notice from the
| 70s whose first sentence is a definition with a bunch of proper
| nouns and whose main theorem is given at the most general
| possible level with no applications at all.
|
| That seems like _exactly_ the thing you want, if you are
| searching for a particular piece of information, the
| typesetting aside.
|
| Especially the generality is important if you actually care
| about the result.
| sublinear wrote:
| Terse statements are easier to prove.
| humansareok1 wrote:
| >whose main theorem is given at the most general possible level
| with no applications at all.
|
| Outside of Applied Math why would this be an expectation at
| all?
| GTP wrote:
| Maybe OP means "application to a concrete example" to help
| the reader understand. If a paper is presenting a theorm,
| seeing it applied to an example could actually help.
| mycologos wrote:
| Yup. I don't mean "here's how you can use my theorem to
| build a bridge", I mean "here's an instantiation used to
| prove a more tangible result". Bonus points if multiple
| such results fall out of the general theorem. That's good
| evidence to me that generality is actually accomplishing
| something.
| lupire wrote:
| Because it's an expository paper, not a reference manual.
| Applications help communicate and create understanding.
| Sirizarry wrote:
| I've known a few very intelligent maths professionals and
| although good people, they always struck me as a bit robotic. I
| know it's anecdotal and a small sample size but I wouldn't be
| surprised if a certain personality is needed to excel and it
| just happens to be very terse and overly professional. I
| however also think that that's a big reason I never got into
| advanced mathematics in the first place. I can't stand terse
| and overly professional material. I get bored much too easily.
| araes wrote:
| > c)
|
| Is there a field of math that's something like "local actors
| put in what appear to be rational choices, yet to external
| observers it often appears 'broken' or 'bad'"? Seems like a
| field of game theory or something. Many times, those internal
| view the situation as acceptable.
|
| Politics in America seems like it is almost always this type of
| result. All local actors, all take rational choices, and all
| America says politics is a ______ (choice of 50 negative words)
| https://www.pewresearch.org/politics/2023/09/19/americans-fe...
| daxfohl wrote:
| The same should be said for engineering design docs tbh
| serf wrote:
| absolutely disagree.
|
| an engineering design document is about information retrieval,
| not education. It's not made to entertain or educate about new
| concepts, it's made to be terse and rigidly structured for the
| sake of aiding the work of the reader and to provide reliable
| information recall methods for those reading it.
|
| I don't want to know what 'problems are worth attacking' when
| reading a design document, I want to know what tolerance
| criteria the hole on the left flange needs to meet.
|
| There is more wiggle-room for artistic expression when we're
| talking about analytical papers like feasibility and failure
| analysis. It doesn't belong in the design phase. It creates
| confusion and ambiguity for the reader often, and those two
| things need not be introduced into design more than they
| already exist.
| Jensson wrote:
| > It's not made to entertain or educate about new concepts,
| it's made to be terse and rigidly structured for the sake of
| aiding the work of the reader and to provide reliable
| information recall methods for those reading it.
|
| That goes for math papers as well. Both needs to be
| understood by juniors, both are used by experts for lookup
| and work.
| eternityforest wrote:
| I might want to know _why_ the hole on the left flange needs
| such insane tolerance though.
|
| Maybe it's not relevant in a specific version, or maybe I
| have an idea for how to solve the issue so it can be made
| cheaper, etc.
|
| Maybe it's absolutely integral to the whole application and I
| should stop wasting my time trying to make a printed PLA
| version, etc.
| constantcrying wrote:
| Absolutely not. Engineering design docs should be terse and to
| the point.
|
| Documentation is not a medium where you want to tell a story,
| because that makes it instantly much less usable, since people
| will read it at random parts to attain specific information.
| vaylian wrote:
| Maybe we have different ideas about what a "story" is, but
| stories are about defining a context and a progression from
| that context to a more advanced state. You can use technology
| in ways that the inventors never intended, but most of the
| time you want to know in which context the technology has
| been developed and which problems it can solve. That is why
| it makes sense to have stories in technical documentation.
| constantcrying wrote:
| >but stories are about defining a context and a progression
| from that context to a more advanced state.
|
| Exactly. Starting a story 80% in makes it nonsensical. If
| your engineering docs are nonsensical if you open them at
| 80% you have failed as a technical writer.
|
| >most of the time you want to know in which context the
| technology has been developed and which problems it can
| solve.
|
| ABSOLUTELY NOT. Imagine your dish washer manual going over
| the history of dish washers interspersed with comments
| about how it functions. That would be insane, useless and
| unreasonable.
| twelfthnight wrote:
| > Ideally the tricks I'm suggesting here will be almost
| invisible, affecting readers in a subliminal way
|
| Why would I want a math paper to be subliminally manipulating me?
| I feel like everyone has been watching too much YouTube/tiktok
| and is buying into the notion that clickbait isn't just a vicious
| feedback cycle destroying everyone's integrity.
| johncarlosbaez wrote:
| Everything is always subliminally affecting you. It might as
| well do it in a helpful way.
| twelfthnight wrote:
| > everything is always subliminally affecting you
|
| Right, but certain methods are more effective than others.
| This paper is arguing and encouraging exactly how to
| manipulate more effectively.
|
| > It might as well do it in a helpful way
|
| Being more effective in teaching I agree is a good thing. But
| a math paper isn't for teaching, it's for showing a proof or
| making an argument. I just think we ought to set standards on
| academic research to remain as neutral as possible to let
| ideas flourish on merit rather than cunning tricks.
|
| EDIT: I get that a career in academia requires all these
| games to get more citations. Looks at ML research, I feel
| like abstracts are written by used car salesman nowadays. So
| like, if you have to do it do it. But we ought to call it out
| from time to time.
| elbear wrote:
| I think it was the word "subliminal" that made you think of
| manipulation.
|
| On the other hand, I read the quote you posted as meeting
| the reader at their level and guiding them to a clearer,
| deeper understanding by providing information in a logical,
| intuitive way. This would mean, for example, providing
| real-world context for each abstract concept introduced,
| rather than just leaving the concept by itself together
| with an abstract definition.
| ajkjk wrote:
| I mean.. It's no different than a story being written better
| instead of worse. The dry paper is just worse in every way, at
| both the author's goals and your goals.
| twelfthnight wrote:
| > This may require "watering down" the results being
| described -- stating corollaries or special cases instead of
| the full theorems in their maximal generality. Sometimes you
| may even need to leave out technical conditions required for
| the results to really be true.
|
| This is a trade off, don't you think? Without the marketing,
| the paper would be more complete and correct.
| lupire wrote:
| No, your exercpt is not complete and correct.
|
| "So, the _introduction_ to a math paper should set the
| scene as simply as possible "
|
| The introduction is not the whole paper.
| twelfthnight wrote:
| If I have a salad and I water down just the dressing the
| whole salad is still worse, no? Unless you are saying the
| introduction isn't important, in which case why would you
| need to change it at all to make the paper more engaging?
| Ar-Curunir wrote:
| No, you skipped the context, which says that in the
| introduction you talk about special cases to build
| intuition, and then provide the general result in the
| technical part.
| twelfthnight wrote:
| Huh, I guess I missed the part about providing the
| general result in the technical part (still don't see it,
| but that makes sense), ultimately that does seem like a
| good idea. At least I've heard we understand better from
| examples first and then generalities.
|
| My (admittedly grumpy) gripe is more that the aim of the
| blog is that "dull" is bad and suggests to add/subtract
| candid mathematics with "heroes" and "conflict". If the
| paper isn't _clear_, that's one thing, but "trick"ing the
| reader to spend more time on your article than they would
| without the embellishment is patronizing at best and
| disingenuous at worst.
| dieselgate wrote:
| Was initially expecting this to be more "elementary education"
| focused rather than academic math. Good article. As a non-
| academic, it is cool to see the idea of "what makes a good paper"
| explored. A lot of the concepts mentioned seem to hold up really
| well in theory but ultimately just seem like stylistic
| differences, to me. Academic writing can have an international
| audience and perhaps "just being technical" has advantages. That
| doesn't change the point of the article, though.
| layman51 wrote:
| This really reminds me of a writing course I took called "Writing
| Stories for Science". That's where I first encountered the idea
| of story beats. The class seemed more focused on the natural
| sciences and story structures, so I got the impression this must
| be very difficult to apply to pure mathematical writing. Maybe
| it's easier in applied mathematics where there's a clearer reason
| that could be explained to a layperson.
| hazbot wrote:
| I don't think the point here is to make a math paper
| understandable to a lay person, but rather make it interesting
| and contextualize it for other academics.
| 082349872349872 wrote:
| Other academics are already interested and either already
| have or know where to get the context; that's why the style
| is the way it is.
| js8 wrote:
| IMHO it's even worse. Once we figure out P vs NP, we will
| understand what it means to invert boolean functions
| algorithmically, and all mathematics will be replaced with an
| automated process.
|
| Today, most of the math is figuring out how to solve an equation,
| i.e. to give an algorithm to calculate the solution. We then
| "quantize" the algorithm to a state machine (e.g. convert reals
| to floats, algorithm steps to machine code), so that we could run
| the state machine on a computer and get a result.
|
| However, once you know what it takes to invert boolean functions,
| you don't need the solution step, you can just quantize your
| equation (problem) directly to a SAT instance, and let the
| inversion algorithm do all the hard work. No (elegant and
| readable) math is required anymore.
|
| So I think we better treat mathematics as a "useless" human
| artifact (like art or chess) rather than something of practical
| value.
| yau8edq12i wrote:
| > IMHO it's even worse. Once we figure out P vs NP, we will
| understand what it means to invert boolean functions
| algorithmically, and all mathematics will be replaced with an
| automated process.
|
| I'm sorry, but that's just naive techno-solutionism. Maybe the
| computer will be able to solve your syntax problems, i.e.,
| write proofs. It will never know anything about the semantics.
|
| Said differently. Maybe the computer will be able to give you a
| proof on demand on statement number 123456789 in some Godel
| numbering of statements. What it won't be able to tell you is
| that the theorem should be called the "intermediate value
| theorem" and what it _means_. Math isn 't just coming up with
| formally correct proofs for pre-existing statements.
|
| I also don't see what "P vs NP" has to do with anything.
|
| > Today, most of the math is figuring out how to solve an
| equation, i.e. to give an algorithm to calculate the solution.
| We then "quantize" the algorithm to a state machine (e.g.
| convert reals to floats, algorithm steps to machine code), so
| that we could run the state machine on a computer and get a
| result.
|
| I'm a mathematician. What you wrote is just bonkers. _Some_
| parts of math are about writing algorithms to solve equations.
| The vast majority of math isn 't.
| js8 wrote:
| My point is, the need for semantics is a human construct,
| it's not required to solve practical problems. Practical
| value of MVT is that we can apply it in the process of
| describing a solution, i.e. an algorithm. But if you have a
| SAT algorithm that can somehow inherently apply a finite
| instance of MVT, you no longer need to know what MVT is.
|
| And this is at the heart of P vs NP question, how to solve
| SAT, and once we understand that, all the math that is about
| solving equations (i.e. that of practical value) will become
| very boring. Kinda like when a game becomes solved with a
| computer.
|
| (This reminds me of Chomsky and Norvig debate on the machine
| learning and nature of understanding. I wouldn't think I
| would take Norvig's position, yet here we are.)
|
| Addendum: I am also not arguing that computers writing proofs
| are a necessity for this, but rather efficiently (as
| possible) solving SAT instances. You don't need to state a
| hypothesis (and thus prove it) for an infinite number of
| cases to solve practical problems.
|
| I am also not judging value of math - yeah, it is fun. All I
| am saying it inherently has a human dimension in the sense we
| don't need it (assuming we have an efficient SAT solver) to
| solve practical problems.
| seanhunter wrote:
| Most equations that can be solved can already be solved
| symbolically by things like mathematica or numerically by a
| grab-bag of numerical techniques that every maths,
| engineering and physics student learns at some point. You
| don't need to have found the solution to P vs NP to do any
| of that.
|
| I think most mathematicians would say that most of maths is
| not in solving equations because once you get the right
| equations, solving them is a simple mechanical process for
| the most part. Most of maths is figuring out what the right
| equations are in the first place and proving that what you
| have done is legitimate.
| js8 wrote:
| I disagree that converting from an equation to a solution
| algorithm is a straightforward process. It is very much
| where mathematics helps, for example just to ensure
| convergence of the method.
|
| Also, the way you model reality (choose the equation) is
| often done with the limitations of solvability in mind.
|
| I also disagree that maths can help you pick the right
| equations, just like there is no formal method that can
| verify whether a program specification is appropriate to
| the problem.
|
| Equation (or a model) is just an infinite family of SAT
| problems, typically parametrized on our understanding
| what a "number" is. If we had a really good SAT solver,
| then getting a solution would indeed be a really
| straightforward process (just pick the number
| representation) of solving the instance. The same is also
| true for checking "legitimacy" (which I interpret as
| consistency) of the model - SAT solver would tell you
| that the instance is unsatisfiable and why. But you can
| never formally determine that the chosen SAT instance is
| appropriate model for the task at hand.
| constantcrying wrote:
| None of this is true and your understanding of the subject
| is obviously very poor.
| js8 wrote:
| Before there was AlphaGo, many top Go players believed
| that understanding Go is somehow special in helping us to
| understand the universe. The AlphaGo was a shock, because
| it turned Go into "just another game", which can be
| beaten by enough calculations on a powerful computer.
|
| I think the mathematics is the same. Lot of its beauty we
| cherish, because we don't properly understand it on the
| "raw" level of SAT (I would write logic, but I really
| mean the simplest metalogic you need), which is evidenced
| by our lack of understanding of P vs NP. Once it is
| understood, the process of "doing math" will be
| understood algorithmically, and it will become "boring"
| or "dull".
|
| But as I said, I don't think lack of universal meaning
| (or lack of mystique) has to detract from the beauty of
| mathematics.
| constantcrying wrote:
| Some facts to consider:
|
| - Polynomial problems can still require enormous
| computing power to the point they are unsolvable. Even if
| tomorrow there is an algorithm transforming every NP
| problem to a P problem, not all problems will be feasibly
| solvable. A O(n^100) Problem is still P.
|
| - P vs. NP is about algorithmic complexity, it has no
| implications about mathematics being an algorithm or not.
|
| - The most likely outcome is that P and NP are different,
| in which case there are zero implications on
| computational efficiency.
|
| - Considering mathematical statements as algorithms has
| happened for a long time. Any outcome of P vs. NP has
| limited implications on the philosophy behind it. It just
| might make it somewhat better.
|
| You just don't understand what P vs. NP is about, your
| posts are bizarre in the actual context of the problem
| statement. The implications on mathematics just aren't
| what you think they are.
| js8 wrote:
| I think what's more fundamental than just answering P vs
| NP, is what exactly makes boolean functions hard to
| (pseudo)inverse? And it's the latter where answering
| (unless the answer comes from an oracle) the former will
| help.
|
| And this question is crucial, because everything we can
| ask computers (or restated, math with real-world
| applications) can be rephrased in terms of finding a
| solution to a SAT instance. It can be a very large
| instance but if there is a practical way of approaching
| the problem, it shouldn't matter.
|
| I think you're assuming that future computer scientists
| and mathematicians will continue to run (many different)
| algorithms, and I don't. I think there might be a single
| algorithm, a SAT solver, which will subsume all others.
| constantcrying wrote:
| Do you not realize that SAT might just be O(2^n) at best?
| Even if by a gigantic miracle SAT is in P, unless there
| is another incredibly great miracle and it is something
| like O(n^3), which would almost certainly be the greatest
| discovery of CS, it would not replace almost any other
| algorithms? Simply because many of the most important
| ones are O(n^2) or even less.
|
| >I think there might be a single algorithm, a SAT solver,
| which will subsume all others.
|
| This is legitimately insane. You might as well hope for
| free energy, FTL Travel, or magic.
| js8 wrote:
| > Do you not realize that SAT might just be O(2^n) at
| best?
|
| Yes, but that's the worst case on all possible instances.
| Even with this worst case performance, there might be an
| algorithm that works on par with other algorithms that
| specialize only on certain instances. E.g. there can be
| an O(2^n) algorithm for general SAT which performs in
| O(n^(3/2)) on XORSAT. We simply don't understand the
| problem well enough to tell.
|
| Also the set of worst-case instances can be vanishingly
| small in practice.
| constantcrying wrote:
| The _average_ case is also O(n^2).
|
| You have a completely utopian idea about what might be
| computationally feasible. I don't even know what you are
| arguing. Certainly it is thinkable that SAT is somehow
| ridiculously simple to calculate, just in the same way
| that achieving simple and near free fusion energy is
| thinkable. It just isn't going to happen and speculating
| on it is pure science fiction.
| samatman wrote:
| > _Certainly it is thinkable that SAT is somehow
| ridiculously simple to calculate, just in the same way
| that achieving simple and near free fusion energy is
| thinkable. It just isn 't going to happen and speculating
| on it is pure science fiction._
|
| These are, as I'm sure you know, utterly different.
| Nuclear fusion is real, 'simple' and 'near free' are
| engineering problems. If human civilization lasts long
| enough, and continues to progress technologically, we'll
| have that, for some value of 'simple' and 'near free'.
|
| On the other hand, the conjecture that SAT is
| algorithmically feasible may be disproven, and in my
| opinion, will be. That would be "just isn't going to
| happen".
| curtainsforus wrote:
| Strange, for you to say this in the age of transformers. They
| are obviously not there yet, but it seems inevitable to me
| that e.g. gpt 'understands' things better than a markov
| chain, and that future systems will understand more.
| js8 wrote:
| I know you're responding to someone else, but funny you say
| that. I suspect modern NN's are just iteratively solving
| some really large instances of SAT (similar to WalkSAT), to
| find a representation of a circuit that best matches a
| function defined on training examples (which are the
| constraints). So they're, unwittingly, kinda the SotA SAT
| solvers.
| constantcrying wrote:
| >IMHO it's even worse. Once we figure out P vs NP, we will
| understand what it means to invert boolean functions
| algorithmically, and all mathematics will be replaced with an
| automated process.
|
| Literal nonsense. Do you not realize that P vs. NP might have a
| negative outcome (extremely likely)?
|
| >Today, most of the math is figuring out how to solve an
| equation, i.e. to give an algorithm to calculate the solution.
|
| Plainly false. In basically any field proving the existence of
| a solution is far, far harder than calculating the solution
| numerically. See e.g. PDEs.
|
| >We then "quantize" the algorithm to a state machine (e.g.
| convert reals to floats, algorithm steps to machine code), so
| that we could run the state machine on a computer and get a
| result.
|
| Literally not true. Not even the assumption is right.
| js8 wrote:
| > Do you not realize that P vs. NP might have a negative
| outcome (extremely likely)
|
| The outcome will matter less than having an answer. And I do
| think it will be a negative outcome either way, because it
| will obsolete lot of beautiful math, and replace it with a
| more mechanical process.
|
| > In basically any field proving the existence of a solution
| is far, far harder than calculating the solution numerically.
|
| First of all, if you want to ascertain whether the numerical
| algorithm works correctly, you have to prove the existence of
| the solution. That's part of the (equation -> solution
| algorithm) conversion process.
|
| Now, you can possibly get by without proving existence of the
| solution for the infinite family (and confront the numerical
| solution directly with reality). But doing that is more
| complicated than, assuming there is a SAT solver, plug the
| SAT instance into it and see if there is a solution (and you
| can get, for a finite family of instances, a similar
| guarantee you would get from the existence proof).
|
| > Not even the assumption is right.
|
| Not sure what assumption you refer to. Computers don't run
| algorithms (because algorithms handle unbounded
| input/output), computers are just a very large state
| machines. So you need to convert the algorithm to a state
| machine first. Saying "run algorithm on a computer" is just a
| useful figure of speech, but somewhat misleading, because it
| implies certain quantization.
| constantcrying wrote:
| >The outcome will matter less than having an answer. And I
| do think it will be a negative outcome either way, because
| it will obsolete lot of beautiful math, and replace it with
| a more mechanical process.
|
| The negative outcome will have exactly zero implications
| about anything. Except "turns out we were right all along"
| polynomial problems are just fundamentally less hard than
| problems you can verify the solution of in polynomial time.
| This has ZERO implications on anyone, except the
| mathematicians who prove it, who will get a price and quite
| a bit of academic fame. That is literally it.
|
| >But doing that is more complicated than, assuming there is
| a SAT solver, plug the SAT instance into it and see if
| there is a solution (and you can get, for a finite family
| of instances, a similar guarantee you would get from the
| existence proof).
|
| Even is SAT is P that doesn't imply a solution can feasibly
| be found. And if P neq NP, then SAT might not be P.
| js8 wrote:
| > Even is SAT is P that doesn't imply a solution can
| feasibly be found.
|
| It does, because in this case you also assumed that
| running numerical algorithm was feasible. So if your SAT
| solver fails to find the answer, it is either suboptimal
| method (because numerical method exists), or (if the
| instance is UNSAT) the numerical method gives you a
| flawed result.
| constantcrying wrote:
| Do you not understand that an O(n^100) algorithm is in P,
| but can never be run feasibly on a computer for any non
| trivial amounts of n? For n=10 you are already far beyond
| the number of Atoms in the universe.
|
| I don't think you have any clue what you are actually
| talking about.
| culebron21 wrote:
| It's not just papers. I tried to learn probabilities theory &
| statistics, deeper than little knowledge I kept from the uni. For
| instance, wanted to understand how you solve problems like
| samples in quality control: if in a sample of N items, m are bad,
| what's the chance X% are bad in production?
|
| Unfortunately, there are either introductory materials (toss a
| coin -- 50% chance faces) or some robot language. Or
| schizophrenic: like starting from the middle of a speech.
| golol wrote:
| I feel like you must be a using a wrong approach to searching
| for mathematics. In your case I would search for a script or
| book with title "Introduction to probabilify theory" or
| "Introduction to mathematical statistics". Then you skim
| through it to see if you can find an analysis of your problem.
| If not, you can hopefully find out what the right keywords are
| to then find a more advanced script or book for the specific
| subfield your problem needs.
| culebron21 wrote:
| That's what I tried to avoid in the first place :)
| KeplerBoy wrote:
| But that's the entry point. Well written textbooks, which
| discuss the right problem are invaluable. If you head
| straight to google scholar, you're going to have a bad
| time.
| constantcrying wrote:
| So you tried to avoid an instructive text which is set up
| to allow readers to learn about the details of the subject
| starting from the ground up? And you are complaining that
| you can't find a resource which does exactly that?
| culebron21 wrote:
| I don't need ground up. I've even passed the exams on
| prob.theory and statistics. I need particular parts of
| it, but not in a cryptic form. I've had read enough of
| textbooks in the uni to see they're just as cryptic as
| research papers.
| constantcrying wrote:
| You can skip chapters you already understand.
|
| I don't know what textbooks you are reading, but almost
| every single one I have read tried very hard to present
| the content in a matter which focuses on understanding,
| unlike papers which focus on pure information.
|
| I am afraid if you find either cryptic you have a serious
| lack in the prerequisite knowledge and that is what you
| need to focus on if you want to understand the subject.
| From first hand experience I can tell you that I have
| passed exams on a lot of things I have very little
| knowledge of right now. Textbooks are essentially the
| only way to reliable self study academic materials.
| constantcrying wrote:
| The solution is to read textbooks. Good textbooks on these
| subjects certainly do exist and you just need to find and read
| them.
| ykonstant wrote:
| Feller's Probability volume 1 is always a good remedy for
| wounded probabilists :) Disregard the complaints that it is
| old-fashioned and dive in.
| _dain_ wrote:
| I find that coding up a Monte Carlo simulation is a tremendous
| help when I have to deal with some probability/statistics
| problem. If I can play with the parameters and immediately see
| how the scatterplot changes, I get a much stronger intuition
| than I could from formal reasoning.
| gumby wrote:
| The author may well be on to something but personally I hate
| "story mode" in popular science books, and would _really_ hate it
| in actual science papers, both the ones I read for work and the
| ones I read for fun. I want to go straight to the equations --
| often I prefer them to the graphs.
|
| But (not joking here) this is a perfect opportunity for an LLM --
| two opportunities, actually.
|
| LLM A takes a dry paper and gives it context. It could make up
| the context but a good one would look up and offer an anecdote
| from Riemann's life or something. I see nothing wrong with that.
|
| And LLM B could take a paper with that stuff, which to me is
| fluff, and strip it all out, leaving the dry bones for me to pick
| over and savour.
|
| It would really just be another form of language translation, if
| a higher level one.
| 082349872349872 wrote:
| I would say "story mode" in papers is like dancing about your
| doctorate: really cool when the combination works, but the
| latter is where all the value lies, so it shouldn't sacrifice
| anything for the former.
|
| What about LLM C, which takes a set of papers as vertices,
| forms an abstract simplex of all their combinations, and then
| spits out new papers on the ten most interesting higher-
| dimensional faces?
| somenameforme wrote:
| This was my initial reaction as well, as I have complete
| disdain for the lowest common denominator approach to many
| things in modern society. But then something occurred to me -
| IMO one of the most well written scientific papers is
| Einstein's special relativity paper. [1] But it's absolutely a
| 'story mode' paper! A moderately educated individual could
| easily understand and follow the paper, even if they might not
| necessarily follow all the math. It just flows inordinately
| better than most modern papers - most of which are written on
| comparably simple and evolutionary (rather than revolutionary)
| topics.
|
| Of course this may be an issue of domain. I'm mostly interested
| in cosmology/astronomy/physics, where math is a tool rather
| than the object of the paper itself.
|
| [1] -
| https://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf
| Someone wrote:
| I don't see this article argument for a "lowest common
| denominator" approach. For example, how is replacing
|
| _"Let M be a complete Riemannian manifold, G a compact Lie
| group and P - M a principal G-bundle."_
|
| by
|
| _"One of the main problems in gauge theory is understanding
| the geometry of the space of solutions of the Yang-Mills
| equations on a Riemannian manifold."_
|
| doing that? It gets rid of "Lie group" and "G-bundle", but
| adds "gauge theory" and "Yang-Mills equations"
|
| Also, "Of course, one should not give a detailed blow-by-blow
| account of every pitfall and wrong turn", IMO, is an argument
| against doing that. It more or less says: "assume that there
| are steps your readers can make on their own".
| sethhochberg wrote:
| As someone who never took much advanced math or physics in
| school and really doesn't understand what either
| representation of that material is staying, personally I
| find the second example far, far more approachable because
| it is far more googleable!
|
| If Gauge Theory is a concept required to relate to the
| other content in that sentence I've got no shot of knowing
| gauge theory is involved in the first example. I don't know
| what the arrow between P and M means. I'd have to look up
| what a G-bundle is. Its basically not clear to me which
| parts are syntax and which are proper nouns.
|
| The example which reads more like prose than an equation
| expressed in terms of English words gives me much, much
| more context for where to begin reading about what I don't
| know.
| gumby wrote:
| One important element for me is that the framing should be
| as specific as possible.
|
| The article's example, regrettably isn't the best example
| of this, but: by saying "one of the main problems..." the
| author (quite reasonably, not click-baiting) is trying to
| say "this paper is about something important and is worth
| reading, or at least reading the abstract". Unfortunately
| _for me_ (not necessarily others) these kinds of context
| act as framing, so I am less likely to match to a similar
| case in a different domain.
|
| Here's a CS example: let's just say you might have found a
| way to, say, compress the TLB or use fewer instructions to
| use it which speeds things up in most cases but slows them
| down in a few corner cases. You could start by talking
| about the problem of paging systems under high load or
| large RAM or something -- great!
|
| But if you described a novel hashing architecture, later in
| the paper pointing out that it's "..useful, for example in
| a pager", I might read the paper and say "holy cow this
| would work well for this thing I'm working on".
|
| That's why I prefer just the dry bones. I can hang whatever
| flesh I want onto them.
|
| But I know not everybody is like that, and that's OK. The
| world isn't supposed to pander just to my need (though it
| should, dammit!)
| gravescale wrote:
| I think a lot of it comes from following advice similar to
| "Write a Catchy First Paragraph" and it goes too far. You end
| up starting out with bizarre barely-sequiturs like "Fiona was a
| graduate student in lower New York in a family cafe run by a
| man named Joao sipping her usual order of single-origin
| cappuccino on a rainy Wednesday" before we even find out what
| the article is about, let alone what the actual insight is.
|
| Furthermore, a lot of popular science ends up using the people
| involved as _the_ lens through which the ideas are eventually
| viewed. Which makes a lot of sense for professional writers who
| are probably more attuned to the human interest than technical
| people. For an example, the first thing I did with a Lego
| vehicle model kit was to throw the little figurine into the
| "junk bits box" and proceed with a now-robotic model. Many
| things are more like _Oppenheimer_ than _Trinity Device
| Annotated Systems Manual_. Which doesn 't mean they're wrong,
| per se: the audience for it is probably bigger and the overall
| "utility" of the work is higher. And even a grump like me knows
| you shouldn't completely ignore human factors. On top of that,
| people who can write the complex technical stuff often don't
| want to mess about in the middle ground. But the bimodality is
| still annoying to me: people-centric "Stories" or deeply-
| involved dessicated technical material that I don't easily
| understand if it's not my field and not so much in-between.
| darby_eight wrote:
| > leaving the dry bones for me to pick over and savour.
|
| How would you understand the relevance the equations have to
| the overarching finding of the paper? Narrative is just as
| important in tying together apriori reasoning as it is in other
| contexts in all but the most trivial findings, and much of
| computer science is not, in fact, apriori, requiring
| argumentation to justify the abductive reasoning within.
| mayd wrote:
| Some possible counterarguments:
|
| 1. Mathematics is a lot more abstract than it used to be.
|
| 2. Mathematics is a lot more specialised than it used to be.
|
| 3. Non-mathematical content is inaccessible to those who don't
| read English.
|
| 4. Space in academic journals is too precious to waste on
| inessential content.
|
| 5. The style is part of a universal mathematical culture so you
| should fit in.
|
| 6. There are many alternative places to publish nontechnical
| academic writing.
| sweezyjeezy wrote:
| > Space in academic journals is too precious to waste on
| inessential content
|
| Not the biggest issue in maths - the arxiv version usually
| won't match the journal version 1:1
| melagonster wrote:
| and they publish on internet, solid copy is not popular
| today.
| jcla1 wrote:
| Regarding you last point: out of interest, what kind of venues
| were you thinking of? Be this personal blogs of said academics,
| just dumping it on a preprint server or actual ("formally
| published") publications?
| Ekaros wrote:
| For last point. Maybe the universities should step up and use
| that massive administration machine they have build for this
| publishing. Just post it on one of their websites. Link to the
| original paper in the prestigious journal.
| lupire wrote:
| #5 being exclusionary to people with different/better
| ideas/practices is not an pillar worth preserving.
| Ar-Curunir wrote:
| "Space in academic journals is precious"
|
| Well thank god we have preprint servers which have no such
| stupid requirements.
| Tutitk wrote:
| The "dull" version is two times smaller and much easier to read.
| Hard pass for me.
|
| Over time it will probably grow into long-long editorial pieces.
| I will propably have to use AI to strip down the story mode.
| seanhunter wrote:
| This is one of those things that is extremely insidious because
| it holds a kernel of truth but the author takes it to a place
| that in my opinion is really unhelpful.
|
| For example, his opening paragraph "One of the main problems..."
| seems fine to me for setting the context, but I would immediately
| want him to follow with "Let ..." and state the proper
| definition. All of the extra fluffiness means I have to do two
| translations - from the fluffy part to the actual maths and then
| back again - each time to understand what he is getting at.
|
| In my opinion, great maths writing is both rigourous and
| engaging. I would use "Calculus" by Michael Spivak as an example.
| It's really lovely to read but also concise and elegant and the
| beauty of (and love for) the maths comes through on every page.
| The author isn't trying to turn it into a short story and it's
| not padded out with additional rhetorical bullshit like "And now
| we come to a key player: the group of deck transformations." That
| sentence makes me want to puke just a little bit.
|
| But all of the above is a matter of opinion. This, for me is a
| _hard nope_ : This may require "watering down"
| the results being described -- stating corollaries or special
| cases instead of the full theorems in their maximal generality.
| Sometimes you may even need to leave out technical conditions
| required for the results to really be true.
|
| I really _really_ hate it when people do shit like this. State
| things properly even if you need to say something like "don't
| worry about x y z condition I put there for now which will be
| explained later". He says you must warn the reader you're doing
| this but basically I think this is just a hard pass from then
| onwards.
|
| Like if you want to give a simpler version of something, you can
| by all means do:
|
| This is known as seanhunter's theorem, which is usually stated
| as, if blah blah blah...
|
| When x is a real number greater than zero this can be simplified
| as follows:
|
| If x is the number of minutes spent in a meeting and p is the
| number of participants, then the expected value of the meeting is
| given by
|
| v= r/sqrt(x^3p^2) r~N(mu, sigma^2)
|
| ... or whatever.
|
| So you give the real version and then the "special case" version
| that is actually useful most of the time. Like when people give
| you Fermat's little theorem[1] and they say a^p is congruent with
| a mod p but that is equivalent to saying if p does not divide a
| then a^(p-1) congruent with 1 (which is the one you're going to
| actually use most of the time).
|
| [1] https://en.wikipedia.org/wiki/Fermat's_little_theorem
| assimpleaspossi wrote:
| Decades ago, I struggled with the start of a math class as a
| young engineering student until one professor, one day, said,
| "It's easy to calculate how many feet of steel you need to get
| from point A to point B but what if you need to calculate the
| number of feet for the curved support under the Eads' bridge?" He
| then proceeded to show how it's done and everything sunk in after
| that.
| hcks wrote:
| No, not everything should be written in the style of a NYT best-
| seller non-fiction, actually
| the_panopticon wrote:
| always a good read on mathematical writing
| https://www.mathematik.uni-marburg.de/~agricola/material/hal...
| zogrodea wrote:
| Some ight appreciate the following short paper, relatedly. A
| quote is extracted below.
|
| https://uhra.herts.ac.uk/bitstream/handle/2299/5831/903260.p...
|
| "We lecturers naturally worry about the content of our lectures
| rather than the emotions we express in giving them. As human
| beings, students respond immediately to the emotive charge, even
| if they do not understand the content. The lecturer may have
| tried to give a balanced account of the debate between X and Y,
| but his preference for Y shines through. When the students come
| to write the essay on the relative merits of X and Y, they know
| where to put their money. The lecturer might try to balance the
| lecture by suppressing his enthusiasm for Y, but this _objective'
| presentation will make a mystery of the whole exercise. The
| students will wonder why they have to sit through all this stuff
| about X and Y when even the lecturer does not seem to care much
| for either of them. The better strategy is for the lecturer to
| plunge into the works of X, reconstruct X's mental world and re-
| enact X's thoughts until he shares some of X's intellectual
| passions. We can be sure that X had intellectual passions, else
| we would not now have the works of X."
| Retr0id wrote:
| I frequently have to read math/cryptography papers as part of my
| research, but I'm neither a mathematician nor a cryptographer,
| which makes things a bit of a slog.
|
| I think this is mostly just down to me not being the target
| audience, but so many papers seem to be more of a "proof that the
| author understood this thing", rather than an attempt to actually
| convey that understanding.
|
| It reminds me of when programmers needlessly optimize or "golf"
| their code - yes, very clever, but now I can't understand what it
| does.
| A_D_E_P_T wrote:
| The best math book I've ever read -- which I think can completely
| transform somebody's appreciation of math -- was William Dunham's
| "Journey Through Genius - The Great Theorems of Mathematics."
|
| What this book did was place mathematics in human and historical
| context. It starts with Hippocrates' Quadrature of the Lune, then
| moves on to Euclid's proof of the Pythagorean theorem, and moves
| along through history all the way down to Euler and Cantor.
|
| I've always thought that the book's format or method is the best
| way to _teach_ mathematics in a general sense. It beats the rote
| practice of formulae out of context, and it simultaneously
| teaches the history of mathematics and science. I 'm always
| gifting parents of school-age children copies of this book.
| kouru225 wrote:
| Putting it on my list
| gthrow12345 wrote:
| My college advisor gifted me a copy when I graduated, and I
| passed it along to one of the best students that I had. Great
| book.
| lapinot wrote:
| > Of the people who see your math paper, 90% will only read the
| title. Of those who read on, 90% will only read the abstract. Of
| those who go still further, 90% will read only the introduction,
| and then quit.
|
| My personal experience is usually quite different. Perhaps i'm
| very weird but i like to think i'm nothing special. I mostly read
| papers when searching for something specific (referral by someone
| in a discussion, searching for a definition, a proof). I almost
| never read the introductions, at least not in my first pass. My
| first pass is usually scanning the outline to search which
| section will contain what i'm searching for and then reading
| that, jumping back and forth between definitions and theorems. I
| usually then read discussion/related work at the end, to read
| about what the authors think about their method, what they like
| or dislike in related papers.
|
| Abstract and introduction i only read when i have done several
| such passes on a paper and i realize i am really interested in
| the thing and need to understand all the details.
|
| I very much hate this "be catchy at the beginning" and its
| extremist instantiation "the quest for reader engagement". Sure
| you should pay attention to your prose and the story you're
| telling. But treating reader of a scientific paper as some busy
| consumer you should captivate is just disrespectful,
| scientifically unethical and probably just coping with current
| organizational problems (proliferation of papers, dilution of
| results, time pressure on reviewers and researchers). Scientific
| literature is technical, its quality should be measured by
| clarity and precision, ease of searching, ease of generalization,
| honesty about tradeoffs. Not by some engagement metric of a
| damned abstract.
| twelfthnight wrote:
| So, marketing is inevitable and necessary, but I have a
| hypothesis that the current Internet is making it worse. For
| example, creators (I'm lumping in researchers with songwriters,
| actors, etc) used to focus on passing the hurdle of getting an
| "elite" power (record company, publisher, University) to
| support them. Once over that hurdle, they specialized in
| creating and left marketing to the elite.
|
| The elites would pressure the creators to do things they
| thought were marketable, but it didn't always work because
| creators had some leverage in negotiation and a small number of
| elites actually cared about making good stuff.
|
| Now, there are fewer gatekeepers, but instead there is an all
| powerful algorithm. Creators all have to do their own marketing
| in addition to creating, and the algorithm can't be negotiated
| with.
|
| So what we wind up with is insipid YouTube thumbnails and
| myriad academic papers with breathless "state of the art"
| claims.
|
| There are tradeoffs, but I do think it's worth noticing how
| effectively we've started to reward creators for marketing
| rather than creating.
| bluenose69 wrote:
| I remember once reading an opinion piece that suggested that most
| papers should trim the "introduction" section greatly, instead
| referring to key review papers or textbook entries. Although I've
| never followed this advice -- I want papers to be accepted, after
| all -- I can see a lot of merit to it.
|
| The idea is to point readers to cohesive and well-cited
| treatments of the foundational material, rather than presenting
| them with a half-hearted _pro forma_ summary that is unlikely to
| be especially insightful.
|
| Fields that follow this scheme would likely accumulate some
| useful review papers that will actually be _read_ , unlike the
| throw-away citations that appear in conventional introductions.
|
| Would this scheme be beneficial to readers? I think so.
|
| But will it take off? This seems unlikely. I read this opinion
| piece perhaps a decade or two ago, and I've not noticed a change
| in academic writing. If anything, the reverse has been true: I
| see more and more introductions that basically rehash
| introductions from other papers. And with LLM tools, this will
| only get worse ... the further the introduction is from the
| author's actual research interest, the higher the likelihood of
| it being irrelevant, puffed-up, or simply wrong.
| datascienced wrote:
| If academic papers were published in HTML with living links on
| the web it would help. Hypertext solves this but are they not
| allowed to use modern (1990s+) technology?
| lupire wrote:
| URLs die. A good citation can be interpreted by technology to
| search and locate the referenced object.
| queuebert wrote:
| This comment below the article from the author cracked me up:
|
| "Part of why this paper took so long to write is that the file
| was called boring.tex."
| seba_dos1 wrote:
| The entirety of math is "human", there's no way for it to neglect
| that "dimension".
|
| (the article's title is "Why Mathematics is Boring")
| chrismorgan wrote:
| Stephen Leacock answered this topic _perfectly_ over a hundred
| years ago in _Moonbeams from the Larger Lunacy_ , chapter six,
| _Education Made Agreeable or the Diversions of a Professor_.
|
| https://www.gutenberg.org/files/4064/4064-h/4064-h.htm#link2...
|
| Minor excerpts to whet your appetite (but seriously, read it,
| it's excellent humour):
|
| > _In the first place I have compounded a blend of modern poetry
| and mathematics, which retains all the romance of the latter and
| loses none of the dry accuracy of the former. Here is an
| example:_ The poem of LORD
| ULLIN'S DAUGHTER expressed as A PROBLEM IN
| TRIGONOMETRY
|
| ...
|
| --***--
|
| > _Here, for example, you have Euclid writing in a perfectly
| prosaic way all in small type such an item as the following:_
|
| > _"A perpendicular is let fall on a line BC so as to bisect it
| at the point C etc., etc.," just as if it were the most ordinary
| occurrence in the world. Every newspaper man will see at once
| that it ought to be set up thus:_
| AWFUL CATASTROPHE PERPENDICULAR FALLS HEADLONG
| ON A GIVEN POINT The Line at C said to be completely
| bisected President of the Line makes Statement
| etc., etc., etc.
| thpl2k3j4324234 wrote:
| Actually, I find this kind of "cheeky" "all-ponies-and-rainbows"
| "woke-American" approach to math almost vomit-inducing (see
| college intro calculus texts).
|
| There is just so_much_language to read.
|
| It feels like I'm a dumb LLM being trained with a deluge of
| pointless data. Some of us (non-native english speakers) much
| rather prefer the old Mir publications - amazingly terse, and
| extremely insightful. Pity Russia's academic excellence pretty
| much went down the drain after the USSR collapse.
|
| The _best_ math book though is SICM - the clarity of exposition
| using MIT-Scheme makes it a great exemplar of pedagogy. Alas,
| never caught on with the plebs.
| Ar-Curunir wrote:
| What exactly is "all-ponies-and-rainbows" about this sentence:
|
| "One of the main problems in gauge theory is understanding the
| geometry of the space of solutions of the Yang-Mills equations
| on a Riemannian manifold."
|
| The author is not proposing to write wordy prose. He is
| proposing to write understandable prose instead of
| incomprehensible pages of equations.
| j2kun wrote:
| TBH I like both. There's a need for food writing and story
| telling, and a need to cut through the fluff and get to the main,
| precise result you need to know. I oscillate between adoring good
| writing and adoring Erdos' 3-page papers that get straight to the
| point.
| Ar-Curunir wrote:
| The same paper should have both: an introductory section that
| lays out the intuition (possibly via special cases), and the
| main technical body which provides the terse technical details.
| gbacon wrote:
| Thank you. This is beautiful.
|
| Related gems from Simon Peyton Jones are below. In the first, he
| also advocates telling a story.
|
| https://www.microsoft.com/en-us/research/uploads/prod/2016/0...
|
| https://www.microsoft.com/en-us/research/uploads/prod/2016/0...
|
| https://simon.peytonjones.org/assets/pdfs/writing-a-proposal...
| lupire wrote:
| Relevant, one of Simon Peyton-Jones's claims to fame is that he
| was too busy researching and publishing world-class research
| with world-class writing and teaching, that he didn't get a
| PhD.
| woopwoop wrote:
| I think mathematics is in a good place with regards to tolerance
| of self-promotion. I do not think that we should put up with
| excessive hype in the name of "humanizing" papers. I do think
| that a lot of mathematicians do not provide enough detail or
| motivation for their arguments. Not necessarily motivation in the
| sense of "why is this important", but motivation in the sense of
| "we are beginning a three page proof. Let me give you a paragraph
| to give you the outline so that you can fill in the details
| yourself, rather than having to read all of the details just to
| reconstruct the outline."
|
| I do have a pet peeve about mathematical exposition. At some
| point, phrases like "obviously" and "it is easy to see" became
| verboten, or at the very least frowned upon. The problem is that
| it didn't become verboten to skip details (this would be
| impossible in general), and those phrases actually do contain
| information. Namely they contain the information that there
| actually is some detail remaining to fill in here. Often in
| papers there will be some missing detail which is not so hard to
| verify, but whose presence is so ghostly in the exposition that I
| think I've missed somewhere where it was stated explicitly, and
| have to go back. I feel like this is the case of someone excising
| an instance of "it is easy to see that" and replacing it with...
| nothing.
| igorbark wrote:
| culture war aside, there are many other more accurate ways to
| say "details omitted for brevity" than "obviously" and "it is
| easy to see that"
|
| this is also something that makes me want a more interactive
| publishing format, though i understand the good reasons to
| stick to the static quo. if it's easy to see, it shouldn't be
| too hard to write out in a collapsible sidebar for those
| interested
| samatman wrote:
| Like everything in mathematics, "obvious" is a term of art.
| Broadly speaking, it refers to a fact, proof, consequence,
| which is necessary for the proof to advance, but which is
| already established elsewhere, so it does not in itself aid
| in understanding the proof being presented.
|
| A proof is either providing a new result, or is proving an
| established result in a new way. Almost always, a proof will
| need other results, in a way that isn't "interesting"
| (another term of art). The point of introducing these results
| as "obvious" is basically to say "here is something which
| isn't proven by the proof I'm presenting, we need it, but
| there's no need to derive it to follow this proof", ideally,
| with a footnote. As language, it is a bit sly: if something
| is obvious in the normal sense, it will be left out.
|
| It's a problem that the modern style is to elide anything
| obvious in this sense, rather than in the sense of "anyone
| who might reasonably read this paper may be expected to know
| this". But labeling these things as obvious isn't meant in
| the sense "if you don't know this, you're stupid or
| uninformed", or in fact "this will be instantly clear as soon
| as I mention it", it's meant to mean "if you were to follow
| the breadcrumbs and check up on the 'obvious' thing, it
| wouldn't help you much in following my proof, so take my word
| for it. Or, y'know, knock yourself out if this step is
| interesting to you".
| araes wrote:
| This might actually be nice, and its probably not that
| difficult to set up in PDFs or similar.
|
| Would be cool, just because you could see that the details
| were actually "omitted for brevity" and not "omitted because
| they're sketchy". And if you rrrreally want to look through
| the details, then they're fairly easily available.
|
| Downside, it might have a chilling effect on papers because
| the scale of writing necessary.
| zer8k wrote:
| > Not necessarily motivation in the sense of "why is this
| important", but motivation in the sense of "we are beginning a
| three page proof. Let me give you a paragraph to give you the
| outline so that you can fill in the details yourself, rather
| than having to read all of the details just to reconstruct the
| outline."
|
| In graduate school this was the most frustrating aspect of
| paper reading (and writing). It makes sense why it exists
| however. Papers on mathematics in particular are laser targeted
| to a particular niche. As the science progresses you need more
| and more bespoke knowledge of previous work to even start the
| paper you're reading. There's an implicit assumption you've
| done your homework, so to speak, and authors likely feel there
| is no need to provide such a summary. Since, of course, if you
| don't have the pre-requisite knowledge the paper isn't targeted
| at you anyway.
|
| Some of it of course is simply a pride thing. There have been
| many times I've felt the lack of exposition was a way to say
| "I'm better than you". I have no evidence this is the case but
| it would not surprise me.
| Warwolt wrote:
| I feel like this entire comment section grossly misunderstands
| what the author means with "story-mode". It's not about actually
| making anything read like fiction, just the order things are
| introduced to the reader.
| jimmar wrote:
| This was the "good" example:
|
| > One of the main problems in gauge theory is understanding the
| geometry of the space of solutions of the Yang-Mills equations on
| a Riemannian manifold.
|
| Perhaps I'm the wrong type of human, but this still does not
| resonate at all.
| tombert wrote:
| I don't know what's objectively "correct", but my favorite CS
| papers are the ones that try and be more entertaining to read.
|
| Stuff like "Cheney on the MTA", or the "Lambda the Ultimate"
| papers are fun to read, while still dumping a lot of interesting
| information. I also think Lamports papers tend to be more fun
| simply because they use more tangible analogies for things rather
| than sticking with formalized mathematics.
|
| I kind of view the overly dry, super-formal math/CS papers to be
| almost a form of gatekeeping. There's a lot of really useful
| information in a lot of papers, but people don't read them
| because they rely on a lot of formalisms and notation that are
| pretty dry to learn about. Sure, _I_ know what a "comonad" and
| "endofunctor" is, and using terms like that can be _useful_ , but
| I also think that it can sometimes be better to take a simpler,
| more grounded approach to things, or at least work with
| metaphors.
| ivanjermakov wrote:
| I feel like math writing shouldn't be written for the general
| audience. It's proffesionals writing for professionals. And only
| then is the job of journalists and pop science writers to
| "translate" it for everyone.
| Ar-Curunir wrote:
| The author (a mathematician) is not proposing that mathematical
| research papers be written for a general audience. He is
| proposing that they be written in a better way for mathematical
| professionals.
| shadowgovt wrote:
| Math is extremely good for precision and conciseness.
|
| It's a _terrible_ language for communicating novel ideas to
| another human being. The amount of context one needs to grasp
| what is being said is enormous.
|
| That's not to say it doesn't have its place. It's more to say
| that it's almost always the case that if you aren't communicating
| with someone in a parallel research space on a mathematical
| topic, you should supplement that communication with some context
| and de-generalization to get the message across.
|
| I think it's about pattern. If your audience is already familiar
| with a pattern and its common properties (matrix mathematics,
| imaginary number mathematics, infinite series, for example), you
| can communicate an idea concisely by providing them an instance
| that fits a pattern and making a small change. But there are way
| too many patterns to just _assume_ the audience knows what
| context we 're in.
|
| To that end, I generally highly recommend the "3Blue1Brown"
| channel on YouTube as a great dive into multiple math topics,
| because the author does a great job of straddling the notational
| representations and the underlying concepts they describe.
| amai wrote:
| It is a long and old tradition in math to explain your findings
| as obscure as possible to make sure that your competitors cannot
| follow you.
|
| "He is like the fox, who effaces his tracks in the sand with his
| tail" (Abel about Gauss)
| https://hsm.stackexchange.com/questions/3610/what-is-the-ori...
| nso95 wrote:
| It seems unreasonable to expect someone to both be an expert in
| mathematics as well be some great story teller
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