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Math writing is dull when it neglects the human dimension

223 点作者 mathgenius大约 1 年前

34 条评论

mycologos大约 1 年前
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.<p>I have talked with math people about why this is, and responses are some combination of<p>a) being concise and being elegant are the same, same for maximum generality&#x2F;abstraction<p>b) the people who should read the paper don&#x27;t need things explained<p>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<p>(Nobody has ever really said c, but my sense is it&#x27;s true. Academic writing has a lot of imitation of style to prove you&#x27;re part of the in-group.)
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woopwoop大约 1 年前
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 &quot;humanizing&quot; 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 &quot;why is this important&quot;, but motivation in the sense of &quot;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.&quot;<p>I do have a pet peeve about mathematical exposition. At some point, phrases like &quot;obviously&quot; and &quot;it is easy to see&quot; became verboten, or at the very least frowned upon. The problem is that it didn&#x27;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&#x27;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 &quot;it is easy to see that&quot; and replacing it with... nothing.
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gumby大约 1 年前
The author may well be on to something but personally I hate &quot;story mode&quot; in popular science books, and would <i>really</i> 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.<p>But (not joking here) this is a perfect opportunity for an LLM -- two opportunities, actually.<p>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&#x27;s life or something. I see nothing wrong with that.<p>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.<p>It would really just be another form of language translation, if a higher level one.
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A_D_E_P_T大约 1 年前
The best math book I&#x27;ve ever read -- which I think can completely transform somebody&#x27;s appreciation of math -- was William Dunham&#x27;s &quot;Journey Through Genius - The Great Theorems of Mathematics.&quot;<p>What this book did was place mathematics in human and historical context. It starts with Hippocrates&#x27; Quadrature of the Lune, then moves on to Euclid&#x27;s proof of the Pythagorean theorem, and moves along through history all the way down to Euler and Cantor.<p>I&#x27;ve always thought that the book&#x27;s format or method is the best way to <i>teach</i> 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&#x27;m always gifting parents of school-age children copies of this book.
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mayd大约 1 年前
Some possible counterarguments:<p>1. Mathematics is a lot more abstract than it used to be.<p>2. Mathematics is a lot more specialised than it used to be.<p>3. Non-mathematical content is inaccessible to those who don&#x27;t read English.<p>4. Space in academic journals is too precious to waste on inessential content.<p>5. The style is part of a universal mathematical culture so you should fit in.<p>6. There are many alternative places to publish nontechnical academic writing.
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twelfthnight大约 1 年前
&gt; Ideally the tricks I’m suggesting here will be almost invisible, affecting readers in a subliminal way<p>Why would I want a math paper to be subliminally manipulating me? I feel like everyone has been watching too much YouTube&#x2F;tiktok and is buying into the notion that clickbait isn&#x27;t just a vicious feedback cycle destroying everyone&#x27;s integrity.
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Brian_K_White大约 1 年前
I always liked Lockhart&#x27;s Lament<p><a href="https:&#x2F;&#x2F;maa.org&#x2F;sites&#x2F;default&#x2F;files&#x2F;pdf&#x2F;devlin&#x2F;LockhartsLament.pdf" rel="nofollow">https:&#x2F;&#x2F;maa.org&#x2F;sites&#x2F;default&#x2F;files&#x2F;pdf&#x2F;devlin&#x2F;LockhartsLame...</a><p>It makes a very different point about teaching, or learning&#x2F;discovering math, not writing about math.
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lapinot大约 1 年前
&gt; 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.<p>My personal experience is usually quite different. Perhaps i&#x27;m very weird but i like to think i&#x27;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&#x27;m searching for and then reading that, jumping back and forth between definitions and theorems. I usually then read discussion&#x2F;related work at the end, to read about what the authors think about their method, what they like or dislike in related papers.<p>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.<p>I very much hate this &quot;be catchy at the beginning&quot; and its extremist instantiation &quot;the quest for reader engagement&quot;. Sure you should pay attention to your prose and the story you&#x27;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.
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bluenose69大约 1 年前
I remember once reading an opinion piece that suggested that most papers should trim the &quot;introduction&quot; section greatly, instead referring to key review papers or textbook entries. Although I&#x27;ve never followed this advice -- I want papers to be accepted, after all -- I can see a lot of merit to it.<p>The idea is to point readers to cohesive and well-cited treatments of the foundational material, rather than presenting them with a half-hearted <i>pro forma</i> summary that is unlikely to be especially insightful.<p>Fields that follow this scheme would likely accumulate some useful review papers that will actually be <i>read</i>, unlike the throw-away citations that appear in conventional introductions.<p>Would this scheme be beneficial to readers? I think so.<p>But will it take off? This seems unlikely. I read this opinion piece perhaps a decade or two ago, and I&#x27;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&#x27;s actual research interest, the higher the likelihood of it being irrelevant, puffed-up, or simply wrong.
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Retr0id大约 1 年前
I frequently have to read math&#x2F;cryptography papers as part of my research, but I&#x27;m neither a mathematician nor a cryptographer, which makes things a bit of a slog.<p>I think this is mostly just down to me not being the target audience, but so many papers seem to be more of a &quot;proof that the author understood this thing&quot;, rather than an attempt to actually convey that understanding.<p>It reminds me of when programmers needlessly optimize or &quot;golf&quot; their code - yes, very clever, but now I can&#x27;t understand what it does.
gbacon大约 1 年前
Thank you. This is beautiful.<p>Related gems from Simon Peyton Jones are below. In the first, he also advocates telling a story.<p><a href="https:&#x2F;&#x2F;www.microsoft.com&#x2F;en-us&#x2F;research&#x2F;uploads&#x2F;prod&#x2F;2016&#x2F;07&#x2F;How-to-write-a-great-research-paper.pdf" rel="nofollow">https:&#x2F;&#x2F;www.microsoft.com&#x2F;en-us&#x2F;research&#x2F;uploads&#x2F;prod&#x2F;2016&#x2F;0...</a><p><a href="https:&#x2F;&#x2F;www.microsoft.com&#x2F;en-us&#x2F;research&#x2F;uploads&#x2F;prod&#x2F;2016&#x2F;07&#x2F;How-to-give-a-great-research-talk.pdf" rel="nofollow">https:&#x2F;&#x2F;www.microsoft.com&#x2F;en-us&#x2F;research&#x2F;uploads&#x2F;prod&#x2F;2016&#x2F;0...</a><p><a href="https:&#x2F;&#x2F;simon.peytonjones.org&#x2F;assets&#x2F;pdfs&#x2F;writing-a-proposal-mar23.pdf" rel="nofollow">https:&#x2F;&#x2F;simon.peytonjones.org&#x2F;assets&#x2F;pdfs&#x2F;writing-a-proposal...</a>
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Warwolt大约 1 年前
I feel like this entire comment section grossly misunderstands what the author means with &quot;story-mode&quot;. It&#x27;s not about actually making anything read like fiction, just the order things are introduced to the reader.
zogrodea大约 1 年前
Some ight appreciate the following short paper, relatedly. A quote is extracted below.<p><a href="https:&#x2F;&#x2F;uhra.herts.ac.uk&#x2F;bitstream&#x2F;handle&#x2F;2299&#x2F;5831&#x2F;903260.pdf" rel="nofollow">https:&#x2F;&#x2F;uhra.herts.ac.uk&#x2F;bitstream&#x2F;handle&#x2F;2299&#x2F;5831&#x2F;903260.p...</a><p>&quot;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.&quot;
dieselgate大约 1 年前
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.
shadowgovt大约 1 年前
Math is extremely good for precision and conciseness.<p>It&#x27;s a <i>terrible</i> language for communicating novel ideas to another human being. The amount of context one needs to grasp what is being said is enormous.<p>That&#x27;s not to say it doesn&#x27;t have its place. It&#x27;s more to say that it&#x27;s almost always the case that if you aren&#x27;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.<p>I think it&#x27;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 <i>assume</i> the audience knows what context we&#x27;re in.<p>To that end, I generally highly recommend the &quot;3Blue1Brown&quot; 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.
Tutitk大约 1 年前
The &quot;dull&quot; version is two times smaller and much easier to read. Hard pass for me.<p>Over time it will probably grow into long-long editorial pieces. I will propably have to use AI to strip down the story mode.
jimmar大约 1 年前
This was the &quot;good&quot; example:<p>&gt; 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.<p>Perhaps I&#x27;m the wrong type of human, but this still does not resonate at all.
chrismorgan大约 1 年前
Stephen Leacock answered this topic <i>perfectly</i> over a hundred years ago in <i>Moonbeams from the Larger Lunacy</i>, chapter six, <i>Education Made Agreeable or the Diversions of a Professor</i>.<p><a href="https:&#x2F;&#x2F;www.gutenberg.org&#x2F;files&#x2F;4064&#x2F;4064-h&#x2F;4064-h.htm#link2H_4_0020" rel="nofollow">https:&#x2F;&#x2F;www.gutenberg.org&#x2F;files&#x2F;4064&#x2F;4064-h&#x2F;4064-h.htm#link2...</a><p>Minor excerpts to whet your appetite (but seriously, read it, it’s excellent humour):<p>&gt; <i>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:</i><p><pre><code> The poem of LORD ULLIN’S DAUGHTER expressed as A PROBLEM IN TRIGONOMETRY </code></pre> …<p>—⁂—<p>&gt; <i>Here, for example, you have Euclid writing in a perfectly prosaic way all in small type such an item as the following:</i><p>&gt; <i>“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:</i><p><pre><code> 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.</code></pre>
culebron21大约 1 年前
It&#x27;s not just papers. I tried to learn probabilities theory &amp; 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&#x27;s the chance X% are bad in production?<p>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.
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assimpleaspossi大约 1 年前
Decades ago, I struggled with the start of a math class as a young engineering student until one professor, one day, said, &quot;It&#x27;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&#x27; bridge?&quot; He then proceeded to show how it&#x27;s done and everything sunk in after that.
tombert大约 1 年前
I don&#x27;t know what&#x27;s objectively &quot;correct&quot;, but my favorite CS papers are the ones that try and be more entertaining to read.<p>Stuff like &quot;Cheney on the MTA&quot;, or the &quot;Lambda the Ultimate&quot; 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.<p>I kind of view the overly dry, super-formal math&#x2F;CS papers to be almost a form of gatekeeping. There&#x27;s a lot of really useful information in a lot of papers, but people don&#x27;t read them because they rely on a lot of formalisms and notation that are pretty dry to learn about. Sure, <i>I</i> know what a &quot;comonad&quot; and &quot;endofunctor&quot; is, and using terms like that can be <i>useful</i>, 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.
hcks大约 1 年前
No, not everything should be written in the style of a NYT best-seller non-fiction, actually
ivanjermakov大约 1 年前
I feel like math writing shouldn&#x27;t be written for the general audience. It&#x27;s proffesionals writing for professionals. And only then is the job of journalists and pop science writers to &quot;translate&quot; it for everyone.
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amai大约 1 年前
It is a long and old tradition in math to explain your findings as obscure as possible to make sure that your competitors cannot follow you.<p>&quot;He is like the fox, who effaces his tracks in the sand with his tail&quot; (Abel about Gauss) <a href="https:&#x2F;&#x2F;hsm.stackexchange.com&#x2F;questions&#x2F;3610&#x2F;what-is-the-original-source-for-abels-quote-about-gausshe-is-like-the-fox-wh" rel="nofollow">https:&#x2F;&#x2F;hsm.stackexchange.com&#x2F;questions&#x2F;3610&#x2F;what-is-the-ori...</a>
seba_dos1大约 1 年前
The entirety of math is &quot;human&quot;, there&#x27;s no way for it to neglect that &quot;dimension&quot;.<p>(the article&#x27;s title is &quot;Why Mathematics is Boring&quot;)
layman51大约 1 年前
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.
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j2kun大约 1 年前
TBH I like both. There&#x27;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&#x27; 3-page papers that get straight to the point.
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queuebert大约 1 年前
This comment below the article from the author cracked me up:<p>&quot;Part of why this paper took so long to write is that the file was called boring.tex.&quot;
nso95大约 1 年前
It seems unreasonable to expect someone to both be an expert in mathematics as well be some great story teller
the_panopticon大约 1 年前
always a good read on mathematical writing <a href="https:&#x2F;&#x2F;www.mathematik.uni-marburg.de&#x2F;~agricola&#x2F;material&#x2F;halmos.pdf" rel="nofollow">https:&#x2F;&#x2F;www.mathematik.uni-marburg.de&#x2F;~agricola&#x2F;material&#x2F;hal...</a>
daxfohl大约 1 年前
The same should be said for engineering design docs tbh
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seanhunter大约 1 年前
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.<p>For example, his opening paragraph &quot;One of the main problems...&quot; seems fine to me for setting the context, but I would immediately want him to follow with &quot;Let ...&quot; 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.<p>In my opinion, great maths writing is both rigourous and engaging. I would use &quot;Calculus&quot; by Michael Spivak as an example. It&#x27;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&#x27;t trying to turn it into a short story and it&#x27;s not padded out with additional rhetorical bullshit like &quot;And now we come to a key player: the group of deck transformations.&quot; That sentence makes me want to puke just a little bit.<p>But all of the above is a matter of opinion. This, for me is a <i>hard nope</i>:<p><pre><code> 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. </code></pre> I really <i>really</i> hate it when people do shit like this. State things properly even if you need to say something like &quot;don&#x27;t worry about x y z condition I put there for now which will be explained later&quot;. He says you must warn the reader you&#x27;re doing this but basically I think this is just a hard pass from then onwards.<p>Like if you want to give a simpler version of something, you can by all means do:<p>This is known as seanhunter&#x27;s theorem, which is usually stated as, if blah blah blah...<p>When x is a real number greater than zero this can be simplified as follows:<p>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<p>v= r&#x2F;sqrt(x^3p^2) r~N(mu, sigma^2)<p>... or whatever.<p>So you give the real version and then the &quot;special case&quot; version that is actually useful most of the time. Like when people give you Fermat&#x27;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&#x27;re going to actually use most of the time).<p>[1] <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Fermat&#x27;s_little_theorem" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Fermat&#x27;s_little_theorem</a>
bedobi大约 1 年前
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.
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js8大约 1 年前
IMHO it&#x27;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.<p>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 &quot;quantize&quot; 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.<p>However, once you know what it takes to invert boolean functions, you don&#x27;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.<p>So I think we better treat mathematics as a &quot;useless&quot; human artifact (like art or chess) rather than something of practical value.
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