The n-Category Café

Very nice. I like how this paper evidently follows its own advice.

One typo I spotted: This may “watering down”.

That title sounds familiar!

Nicely put!

One thing I think is often under-appreciated about the “mathematical writing as story” paradigm is that, just as in fiction, the plot of the story itself is not the same as the history of how the author came to invent it. An author creating fiction will often have an idea for the climax or some other part of the story first, and only later realize how to lead up to it. They may even jump around during the actual writing of the text, rather than writing from first page to last page in order. I think the same is true for mathematical writing: “telling a story” doesn’t necessarily mean telling the story of how you came to invent/discover and prove your results. The two are probably more closely related than in fiction writing, but I think it’s still important to keep them separate.

Makes sense to have ChatGPT tailor papers to audidence.

Thank you! I have been struggling with writing up a rather long document (currently about 45 pages) and I’ve been worried that no one will read it. I am excited about the possibility of rewriting the introduction with your advice.

I am addicted to reading science fiction, so I love the idea of visualizing my theorems and lemmas as tools or characters fighting the enemy (ignorance is the enemy). I may even try to incorporate these literary ideas into teaching. I haven’t taught for years, but I still feel the desire to teach.

Thanks again. :)

I really do think there is a second reason why some papers are boring, and it goes beyond simply bad writing: the career game of mathematics promotes aggressive publishing, so there is enormous pressure to publish things that simply aren’t interesting, and wouldn’t be even if Hemingway himself wrote them (and was a good mathematician).

I know this is going against the grain here, but instead of focusing on improving the writing of new papers, math departments/universities/etc. should encourage the rewriting of existing older math to an even higher, more interesting standard. Speaking as someone with a PhD in math and someone who had to read a lot of papers, many of them are rather poorly written and simply fail to make the paper interesting. (There are definitely exceptions, obviously. I’ve enjoyed very much the papers of Irving Kaplansky.)

Yes, sometimes people do that in books when a subject becomes cohesive enough I guess, but it’s not really a lot, nor is it an encouraged activity.

Unfortunately, there is an imposed pace upon mathematicial research, the natural consequence of which is simply bad writing that comes about simply because it’s hard to make something good out of a result whose only saving grace is its factual correctness.

There has been great work done in applied linguistics and rhetoric studies on the moves that authors make in various academic genres. The big name in this area is Jon Swales. (Look him up, he has lots of publications and most of them do a good job of communicating his ideas and how they come out of the corpus he has put together.) In particular, he has a model of how experienced, but not necessarily exceptional, academic writers tend to write introductions to articles, called the CARS model (Creating A Research Space).

A question that I think is under-examined is how the CARS model maps onto contemporary models of narrative writing, beyond the very simple sketches that I got in high school English. One of these days a mathematician should team up with someone who teaches in a literature MFA and go to town on this. (I’m a mathematician, not a linguist or rhetorician and definitely not an MFA instructor, but I worked at my university’s writing centre for a couple years, mostly teaching grad students in the exact sciences how to work with models like CARS.)

On Mathstodon Steve Dodge pointed out Randy Olson’s book Houston, We Have a Narrative. Here’s a blurb:

Ask a scientist about Hollywood, and you’ll probably get eye rolls. But ask someone in Hollywood about science, and they’ll see dollar signs: moviemakers know that science can be the source of great stories, with all the drama and action that blockbusters require.

That’s a huge mistake, says Randy Olson: Hollywood has a lot to teach scientists about how to tell a story—and, ultimately, how to do science better. With Houston, We Have a Narrative, he lays out a stunningly simple method for turning the dull into the dramatic. Drawing on his unique background, which saw him leave his job as a working scientist to launch a career as a filmmaker, Olson first diagnoses the problem: When scientists tell us about their work, they pile one moment and one detail atop another moment and another detail—a stultifying procession of “and, and, and.” What we need instead is an understanding of the basic elements of story, the narrative structures that our brains are all but hardwired to look for—which Olson boils down, brilliantly, to “And, But, Therefore,” or ABT. At a stroke, the ABT approach introduces momentum (“And”), conflict (“But”), and resolution (“Therefore”)—the fundamental building blocks of story. As Olson has shown by leading countless workshops worldwide, when scientists’ eyes are opened to ABT, the effect is staggering: suddenly, they’re not just talking about their work—they’re telling stories about it. And audiences are captivated.

Written with an uncommon verve and enthusiasm, and built on principles that are applicable to fields far beyond science, Houston, We Have a Narrative has the power to transform the way science is understood and appreciated, and ultimately how it’s done.

I absolutely agree with your thoughts here. I ran an elementary school math camp for six summers. One thing I learned was that telling a story was a very effective way to teach concepts (sometimes without them even being aware they were learning). I translated that story telling into a popular book series called The Math Kids, where elementary school kids use their math skills to solve mysteries.

My thoughts are:

1. Try to write short papers. I’ve seen papers over 100 pgs or even much more. Thats a book, not a paper. Maybe this highlights problems in the scientific publishing process.

2. Don’t use long strings of adjectives. My currrent bug-bear is Locally convex Hausdorff topological vector space.

3. Don’t use symbols unnecessarily.

4. Use names for mathematical objects that relate to the property under consideration.of course this isn’t eaay because of tradition.

A couple of other of my bug bears in mathematical prose is the use of mathematical jargon where an ordinary English word will do. Like trivial or nontrivial. A trivial result could just as well be an easy or simple or strightfoward result. A non-trivial result can be a hard, difficult or deep result. And repeating tired jokes like “by abstract nonsense” or “abuse of notation” (Can we think of new ones?)

A physical example is saying Hilbert space is an abbreviation for quantum state space. The former is a mathematical object and the latter is physical. We should foreground the essence of the subject under discussion and not the model.

There is surely an obvious point that has not been made yet: mathematical writing is trying to serve two fundamentally irreconcilable goals, namely verification of correctness, and conveying of ideas. Today, as has been the case for more than a century, the first of these two factors is entirely dominant. This is ironic, as any experience with writing computer programs will confirm that they are never bug-free when they are first served up; one hopes there are only inconsequential flaws, but not infrequently some significant tweaks are necessary, and not rarely the whole approach needs adjusting. Human review helps only slightly: it is actually testing (in various ways) and running the code that reveals the errors. One is trying to make mathematics more computer-like, without any computer to run it on.

Verification of correctness of a proof will be easier the more carefully and precisely the steps are laid out. This goes directly against ‘readability’ in the sense of conveying ideas: the paper will be lengthier, more tedious, almost certainly harder to publish, and will only reward someone minutely working through some part of it that they really wish to understand the details of. No amount of expositional flourish is going to help much in this regard, it is simply a matter of clarity and effort.

The dominance of an impossible pursuit of correctness has vast detrimental consequences. It leads to elitism (in the absence of any way for anybody to easily check a proof, one relies on the conferment of acceptance; the more significant and/or difficult a paper is, the more reliant one is on the willingness of the elite in the field to engage with it), it rewards certain types of personality to the exclusion of others, and it stimies creativity.

If one insists on forcing everything into an extraordinarily narrow pronouncement of right or wrong (the notion of which at any point of history is highly fragile: the myth that a mathematical result once obtained is immortal is blinding and damaging psychologically, it is no more so than anything else), democratisation of mathematics and a leap forward in exposition will only happen if one can delegate the proof-checking to a machine, so that a paper will have two parts, the formal and the informal, and a new style of writing can emerge in the informal in which there is no need to give details.

If one were willing to change one’s standards of rigour, one might be able to make progress in other ways. Computer programs are almost never tested in something akin to today’s attempts at mathematical proof checkers, instead one settles for ‘unit tests’ (formalised tests of representative and corner cases), for type-checking of a limited kind, for running on examples, etc. Something like this would be possible for mathematical statements in lieu of fully formalised proofs, and would certainly provide a higher degree of certainty of correctness than a purely human-written argument. There is something a little akin in this to the points of view that Doron Zeilberger has expressed for a long time.

Even better, if entirely inconceivable, would be a turn towards ideas-as-primary. Brouwer’s philosophy on intuitionism has something in common with this: ultimately, the way in which many people truly obtain understanding of a piece of mathematics is in re-creating the proof in their mind in a way which makes sense to them. Their internal conception might not actually be fully rigorous, it might not even entirely make sense, but they believe it in the context of the other pieces of mathematics which they have internalised. One could write mathematics in a way which primarily aims to serve this purpose: rigorous proof as understood today is of very little relevance to this. Sometimes, of course, one might need a piece of mathematics in ‘real-life’ where one really needs details/rigour, or maybe somebody just wishes to give such details, and such an endeavour is entirely valid and praiseworthy, but this is more akin to engineering; there is no need to insist upon it as part of mathematics itself as a creative endeavour.

More discussion over on HN.

“ The reader’s interest a delicate thing.” Missing is?

I always associated the quote you referenced for Piet Hein: “Problems worthy of attack prove their worth by fighting back.” to Paul Erdős (with the slight variant “A problem worthy of attack, proves its worth by fighting back!”).

Do you know who said it first? Perhaps the Erdős quote is mis-attributed, but I do see it in many places.