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.)
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".
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.
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.
> 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?
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.
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 שַשֶׁ!).
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.
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.
> 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.
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
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.
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.
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?
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.
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"
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.
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.
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.
>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.
While I don't think it's practical to expect every paper author to write in the most consise and elegant style possible, one thing I do wonder is the lack of a "comment section" for papers: why isn't there a centralized place for academics to just leave comments or their understanding of the paper, like most of the web forums? I don't expect the author to respond or even be active in every comment section but there should certainly be such a place. Every other form of Internet-based discussion, including Reddit, any random Internet forum, mail lists, Discord, etc. are all interactive and everyone is expected to participate in a centralized and public discussion.
This way, even if the paper itself is hard to read, for the good ones I do expect someone else to leave a comment with a sketch of the most interesting parts of the paper that would get upvoted to the top, and the author could click a "promote" button or something to make it official. This way the author doesn't have to write very well as long as the paper is valuable enough for someone else to be interested.
Any comments section would require moderation, and internet moderation is a huge pain in the butt. It's not an unsolvable problem, as the examples you give demonstrate, but it does require a lot of effort, and I think you'd be pretty hard-pressed to find academics willing to expend that effort.
Personally, I rather like these these; they have a certain retro-appeal, in particular old Springer mathematics publications. We are so spoilt with LaTex.
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...
That's generally referred to as a Prisoner's Dilemma situation, where locally the incentives are to do something that makes it worse for everyone. The SlateStarCodex author generalized the concept into Moloch:
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.
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.
> 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.
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
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.
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".
As a working mathematician, I would say "obvious" is most often used for omitted arguments whose analogues are widely established elsewhere.
As an elementary example, in an analytic number theory paper one might need to know that for some constant C we have
5x + x^3 + ln(x) < Ce^x
for all positive x. (In practice, one would not need to determine such a C, just know that one exists.)
Such a claim would usually be described as "obvious" or else stated without any justification. I doubt you could find precisely this claim in the literature, but one can find plenty of very similar arguments in textbooks, and anyone doing research in analytic number theory would find this "obvious".
> 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.
"Trivial" is a term of art (and doesn't mean exactly this) but I'm not sure that "obvious" is.
I think that something already proved is called an immediate corollary if it's a very small step, theorem if it's not that small but well known, a previous result (with a citation!) if not well known, or if you can't cite anything, at least say it's part of "mathematical folklore".
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.
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.
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”.
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.
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!)
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 João 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.
> 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.
> 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.
I think John Baez agrees with you. At the conclusion he says,
> The ideas here take practice to implement well, and they should not be overdone. I’m certainly not saying that a good math paper should remind readers of a story. Ideally the tricks I’m suggesting here will be almost invisible, affecting readers in a subliminal way: they will merely feel that that paper is interesting, carrying them in a natural flow from the title to the conclusion.
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?
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.
Absolutely my favorite pop-sci/pop-math book of all time, if you can call it that.
That's because it's the only book I know which is in a good sweet spot between being a true pop-math book, giving the history and context of math (kind of like, say, Fermat's Enigma by Simon Singh), but while also being a real math book, and actually teaching real maths and real proofs of all the theorems talked about. There are some similar books, but most don't get the mix right, and even the ones that do, are just not as good.
Such a wonderful wonderful book. Do you have any other recommendations for similar books?
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.
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?
> 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.
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.
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.
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.
> 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.
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?
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.
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.
> 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.
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.
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.
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?
As the meme says: why not have both, or even an extension to HTML for a permanent reference that your browser can then resolve using the least defunkt current resolver.
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.
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.
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.
"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."
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.
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.
> 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.
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.
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.
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.
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.
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.
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?
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
>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.
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.
> 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.
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).
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.
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.
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.”
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.
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.
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.
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.
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.
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.
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.
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.
I am in education and what you claim here is in contradiction with both my experience and the direct feedback of the people I educated — most of which would say they dislike math or say they are bad at it. The argument that it is like it is for a reason can be made, but it has probably more to do with the historical role of school as an institution that is meant to produce obidient workers than with school as a place that is meant to produce people able to analyze nature and the world around them with the tools of mathematics. Ever student that got a bad grade because they solved the problem, but did so using the "wrong" way can testament to this.
Back when I was in school I had the luck to also have a lot of maths both physics and mechanics classes, so I had above average exposure to mathematical applications. This is what made me the most wanted maths-helper among my friends who went to other schools with a "purely theoretical"-maths approach: I could finally show them how the stuff they had a hard time with connects to the actual real world and nature around them — which contrary to your statement really their ability to grasp it.
If we think about how one would go about teaching any subject in a way that creates the maximum disinterest in the field it would be done dry theory only while keeping real world connections and story telling to a minimum. Sure, you could do that and there would always be a small number of students who would excell at that anyways because their brains are wired for abstract things — but that isn't a healthy approach to to take in general education.
Sometimes my feeling is that mathematicians, like many other nerds want to gatekeep and shroud their own field in an overly opaque mysticism, because they can derive a higher self value from that. So things get explained over complicated and they tell you, that you would never understand (because you are the wrong kind of person).
I like to instead believe that if explained right you can nearly explain everything to everyone — provided you manage to spark their interest, keep their interest and provide a good way into the matter. This is why Feynmanns lectures are so famous. He was really good at this.
Feynman’s lectures demonstrate my point. From Wikipedia on the lectures:
As a result, some physics students find the lectures more valuable after they have obtained a good grasp of physics by studying more traditional texts, and the books are sometimes seen as more helpful for teachers than for students.[5]
It does not sound to me that you have much experience with actual teaching.
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
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.
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?
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.
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?
> 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.
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.
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.
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".
No it didn't. I am very confident it did not. All students that make it past grade school go through nearly the exact same sequence and motivating examples beginning from geometry, high school algebra, and then some introduction to calculus if not calculus outright.
And the motivating example has always been how to calculate area, and if advanced enough, volume, for all kinds of shapes, regular or otherwise. This is not a recent development, its been the motivating example and the entire purpose for calculus having been invented.
I am so confident in this that I am here to call you a terrible student and that, yes, you likely were a terrible student, especially since your characterization is the exact standard excuse I hear from terrible students.
So now I have to drill - what was "nonsensical"? What had "zero context"? You immediately have lost the claim of "zero application", so we don't have to go there.
If your claim is beyond K-12 math and into university, then I would be too upset to even argue with you if that is your actual position. Making these claims after more than a decade of learning math tells me you have trained yourself to forget all you learned.
Interesting form of confidence. I didn't write the comment you replied to, but I wrote the original comment.
The first fact I want to have written down is that there are bad math teachers. And sadly it is not uncommon for them to be bad, as I mentioned most of my collegues from other classes and schools had bad math teachers and some of them were people that liked math.
Now as an educator I am aware that students are quick to blame other people or the system for their own deficiencies, but I can tell you for a fact that our math teacher failed to explain us what an integral was actually for in an parsable way, even after multiple times of asking. She just knew how to do the formula stuff, she never used it herself on real applications. And when she did explain things, she always did it once, much too briefly in the beginning, so that by the point at which we started doing her always decidedly abstract examples nobody knew what the fuck this was for. As I said, our physics and mechanica teachers were better math teachers than her. As for the integral I had to go and look it up myself, the internet was still young and there weren't many grown ups who knew how to look such a thing up and when I saw the explainations I wondered why nobody ever told us..?
I understand btw. why it can be an advantage to understand math in purely abstract ways, but in her case she just didn't want to talk much and stick to the examples. And that is bad teaching. Ah and before we get into a discussion about specifics, like many on this site I grew up and live happily outside of the US, so educational plans etc. might differ.
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.
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.
> 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 Gödel 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.
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.
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.
- 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.
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.
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.
> 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.
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.
> 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".
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.
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.
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.
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.
>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.
> 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.
>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.
> 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.
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.
Small correction: If P != NP, then SAT definitely isn't in P. That's because SAT is NP-complete, so if it was in P, every other NP problem could be polynomially reduced to it and would hence also be in P.
I think you miss the point. I am not disputing FOL being undecidable. What I am saying is that, once we know how to solve SAT well enough, most mathematics will become obsolete (especially for applications). It will be obsoleted the similar way that symbolic solving of equations got obsoleted with numerical solving, once computers (and numerical algorithms) became prevalent enough.
Practically minded people will simply skip logical proofs and model the world directly with SAT instances. Mathematics will remain an intellectual curiosity.
In a way, I agree with the article - what makes math interesting is the human perspective and creativity. But I believe it's "worse" - taken objectively, math is actually pretty boring and there is only little depth.
Again, you miss the point, there will be a paradigm shift. Yes, symbolic maths still exists, but nobody is looking up a special function to solve an equation. They just plug the numbers (and the state machine approximation of the numerical operations) in the computer and numerically, the result comes out.
Symbolic/numeric is just analogy of the paradigm shift that will come to all (applied) math. Another example is in statistics, shift from parametric to non-parametric.
Cryptography.. relies on unproven belief in ETH or P/=NP or some such (in fact, cryptography has, since Caesar, relied on unproven beliefs about cryptosystems). But that's just a belief, we don't really understand how hard any of these specific problems are. Which is actually a great example of the new paradigm I am talking about, because it shows we don't need proofs (of statements about infinitely many things). We can just quantize things directly (into a state machine that actually does the crypto stuff) and be happy.
We haven't even scratched the surface of what is possible in P vs NP. In particular, CNF representation of SAT instances might be a really crappy one.
You should look up provable security. And also, while security proofs typically rely on unproven assumptions, correctness proofs in cryptography are "regular" proofs for all intents and purposes. You can't even do RSA etc. without understanding prime numbers which is impossible if you limit yourself to propositional calculus ("SAT").
> Again, you miss the point, there will be a paradigm shift. Yes, symbolic maths still exists, but nobody is looking up a special function to solve an equation. They just plug the numbers (and the state machine approximation of the numerical operations) in the computer and numerically, the result comes out.
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.)
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".
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.
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.
Are these shared / mentioned anywhere?
A resource like this would be super helpful.
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?
[1]: https://en.wikipedia.org/wiki/Nicolas_Bourbaki
[2]: https://press.princeton.edu/books/hardcover/9780691161730/cr...
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 שַשֶׁ!).
If the journal demands you remove the appendix or supplementary, just remove it from the published manuscript. Then add it to your ArXiv submission.
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.
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.
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.
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.
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
Outside of Applied Math why would this be an expectation 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.
This way, even if the paper itself is hard to read, for the good ones I do expect someone else to leave a comment with a sketch of the most interesting parts of the paper that would get upvoted to the top, and the author could click a "promote" button or something to make it official. This way the author doesn't have to write very well as long as the paper is valuable enough for someone else to be interested.
Personally, I rather like these these; they have a certain retro-appeal, in particular old Springer mathematics publications. We are so spoilt with LaTex.
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...
https://slatestarcodex.com/2014/07/30/meditations-on-moloch/
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.
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.
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
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.
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".
As an elementary example, in an analytic number theory paper one might need to know that for some constant C we have
5x + x^3 + ln(x) < Ce^x
for all positive x. (In practice, one would not need to determine such a C, just know that one exists.)
Such a claim would usually be described as "obvious" or else stated without any justification. I doubt you could find precisely this claim in the literature, but one can find plenty of very similar arguments in textbooks, and anyone doing research in analytic number theory would find this "obvious".
"Trivial" is a term of art (and doesn't mean exactly this) but I'm not sure that "obvious" is.
I think that something already proved is called an immediate corollary if it's a very small step, theorem if it's not that small but well known, a previous result (with a citation!) if not well known, or if you can't cite anything, at least say it's part of "mathematical folklore".
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.
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
“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”.
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.
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!)
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.
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.
I think John Baez agrees with you. At the conclusion he says,
> The ideas here take practice to implement well, and they should not be overdone. I’m certainly not saying that a good math paper should remind readers of a story. Ideally the tricks I’m suggesting here will be almost invisible, affecting readers in a subliminal way: they will merely feel that that paper is interesting, carrying them in a natural flow from the title to the conclusion.
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?
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.
That's because it's the only book I know which is in a good sweet spot between being a true pop-math book, giving the history and context of math (kind of like, say, Fermat's Enigma by Simon Singh), but while also being a real math book, and actually teaching real maths and real proofs of all the theorems talked about. There are some similar books, but most don't get the mix right, and even the ones that do, are just not as good.
Such a wonderful wonderful book. Do you have any other recommendations for similar books?
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.
Not the biggest issue in maths - the arxiv version usually won't match the journal version 1:1
Well thank god we have preprint servers which have no such stupid requirements.
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.
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.
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.
This is a trade off, don't you think? Without the marketing, the paper would be more 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.
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.
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.
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.
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.
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.
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.
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...
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."
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.
Over time it will probably grow into long-long editorial pieces. I will propably have to use AI to strip down the story mode.
> 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.
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:
…—⁂—
> 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:
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.
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.
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.
"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...
(the article's title is "Why Mathematics is Boring")
"Part of why this paper took so long to write is that the file was called boring.tex."
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.
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.
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.
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.
That goes for math papers as well. Both needs to be understood by juniors, both are used by experts for lookup and work.
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:
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
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.”
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.
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.
It sounds to me like you haven’t taught much. I could be wrong.
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.
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.
Back when I was in school I had the luck to also have a lot of maths both physics and mechanics classes, so I had above average exposure to mathematical applications. This is what made me the most wanted maths-helper among my friends who went to other schools with a "purely theoretical"-maths approach: I could finally show them how the stuff they had a hard time with connects to the actual real world and nature around them — which contrary to your statement really their ability to grasp it.
If we think about how one would go about teaching any subject in a way that creates the maximum disinterest in the field it would be done dry theory only while keeping real world connections and story telling to a minimum. Sure, you could do that and there would always be a small number of students who would excell at that anyways because their brains are wired for abstract things — but that isn't a healthy approach to to take in general education.
Sometimes my feeling is that mathematicians, like many other nerds want to gatekeep and shroud their own field in an overly opaque mysticism, because they can derive a higher self value from that. So things get explained over complicated and they tell you, that you would never understand (because you are the wrong kind of person).
I like to instead believe that if explained right you can nearly explain everything to everyone — provided you manage to spark their interest, keep their interest and provide a good way into the matter. This is why Feynmanns lectures are so famous. He was really good at this.
As a result, some physics students find the lectures more valuable after they have obtained a good grasp of physics by studying more traditional texts, and the books are sometimes seen as more helpful for teachers than for students.[5]
It does not sound to me that you have much experience with actual teaching.
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.
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?
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?
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.
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.
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.
No it didn't. I am very confident it did not. All students that make it past grade school go through nearly the exact same sequence and motivating examples beginning from geometry, high school algebra, and then some introduction to calculus if not calculus outright.
And the motivating example has always been how to calculate area, and if advanced enough, volume, for all kinds of shapes, regular or otherwise. This is not a recent development, its been the motivating example and the entire purpose for calculus having been invented.
I am so confident in this that I am here to call you a terrible student and that, yes, you likely were a terrible student, especially since your characterization is the exact standard excuse I hear from terrible students.
So now I have to drill - what was "nonsensical"? What had "zero context"? You immediately have lost the claim of "zero application", so we don't have to go there.
If your claim is beyond K-12 math and into university, then I would be too upset to even argue with you if that is your actual position. Making these claims after more than a decade of learning math tells me you have trained yourself to forget all you learned.
The first fact I want to have written down is that there are bad math teachers. And sadly it is not uncommon for them to be bad, as I mentioned most of my collegues from other classes and schools had bad math teachers and some of them were people that liked math.
Now as an educator I am aware that students are quick to blame other people or the system for their own deficiencies, but I can tell you for a fact that our math teacher failed to explain us what an integral was actually for in an parsable way, even after multiple times of asking. She just knew how to do the formula stuff, she never used it herself on real applications. And when she did explain things, she always did it once, much too briefly in the beginning, so that by the point at which we started doing her always decidedly abstract examples nobody knew what the fuck this was for. As I said, our physics and mechanica teachers were better math teachers than her. As for the integral I had to go and look it up myself, the internet was still young and there weren't many grown ups who knew how to look such a thing up and when I saw the explainations I wondered why nobody ever told us..?
I understand btw. why it can be an advantage to understand math in purely abstract ways, but in her case she just didn't want to talk much and stick to the examples. And that is bad teaching. Ah and before we get into a discussion about specifics, like many on this site I grew up and live happily outside of the US, so educational plans etc. might differ.
"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.
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.
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 Gödel 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.
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.
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.
- 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.
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.
SAT is limited to propositional calculus which is far too weak for most mathematics.
>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.
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.
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.
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".
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.
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.
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.
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.
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.
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.
I don't think you have any clue what you are actually talking about.
Small correction: If P != NP, then SAT definitely isn't in P. That's because SAT is NP-complete, so if it was in P, every other NP problem could be polynomially reduced to it and would hence also be in P.
> all mathematics will be replaced with an automated process
FOL is undecidable, so we know this is impossible.
Practically minded people will simply skip logical proofs and model the world directly with SAT instances. Mathematics will remain an intellectual curiosity.
In a way, I agree with the article - what makes math interesting is the human perspective and creativity. But I believe it's "worse" - taken objectively, math is actually pretty boring and there is only little depth.
That never happened, symbolic and numerical mathematics coexist to this day. Cryptography, for example, relies on exact arithmetic.
On that note, good luck trying to solve problems in cryptography with "SAT instances".
Symbolic/numeric is just analogy of the paradigm shift that will come to all (applied) math. Another example is in statistics, shift from parametric to non-parametric.
Cryptography.. relies on unproven belief in ETH or P/=NP or some such (in fact, cryptography has, since Caesar, relied on unproven beliefs about cryptosystems). But that's just a belief, we don't really understand how hard any of these specific problems are. Which is actually a great example of the new paradigm I am talking about, because it shows we don't need proofs (of statements about infinitely many things). We can just quantize things directly (into a state machine that actually does the crypto stuff) and be happy.
We haven't even scratched the surface of what is possible in P vs NP. In particular, CNF representation of SAT instances might be a really crappy one.
> Again, you miss the point, there will be a paradigm shift. Yes, symbolic maths still exists, but nobody is looking up a special function to solve an equation. They just plug the numbers (and the state machine approximation of the numerical operations) in the computer and numerically, the result comes out.
This is false.