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March 28, 2024

Why Mathematics is Boring

Posted by John Baez

I’m writing a short article with some thoughts on how to write math papers, with a provocative title. It’s due very soon, so if you have any thoughts about this draft I’d like to hear them soon!

Why Mathematics is Boring

I don’t really think mathematics is boring. I hope you don’t either. But I can’t count the number of times I’ve launched into reading a math paper, dewy-eyed and eager to learn, only to have my enthusiasm slowly but remorselessly crushed by pages and pages of bad writing. There are many ways math writing can be bad. But here I want to focus on just one: it can be dull. This happens when it neglects the human dimension.

The reader’s interest a delicate thing. It can die at any moment. But properly fed, and encouraged, it can grow to a powerful force. Clarity, well-organized prose, saying just enough at just the right time — these are tremendously important. You can learn these virtues from good math writers. But it also makes sense to look to people whose whole business is keeping us interested: story-tellers.

Everyone loves a good story. We have been telling and listening to stories for untold millennia. Stories are one of our basic ways of understanding the world. I believe that when we read a piece of mathematics, part of us is reading it as a highly refined and sublimated sort of story, with characters and a plot, conflict and resolution.

If this is true, maybe we should consider some tips for short story writers, taken from a typical online guide [K] and see how they can be applied — in transmuted form — to the writing of mathematics. These tips may sound a bit crass. But they go straight to the heart of what gets people interested, and keeps them interested.

Write a Catchy First Paragraph

We are constantly encountering pieces of writing; we don’t bother to finish reading most of them. Once books were rare and precious. Now there is always too much to read. We efficiently cull out most of the material vying for our attention. Often we base our decision on the first sentence or two. Thus, writers of short stories learn the importance of quickly grabbing the reader’s attention. In a catchy story, each sentence makes the reader want to read the next. The first few sentences bear the brunt of this responsibility.

I like to put it this way. 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. Thus, it pays to put a huge amount of energy into making the front end of your paper clear and enticing. This can reduce those 90% figures (which I made up) to about 80%, leading ultimately to an eightfold increase in the number of people who read beyond your paper’s introduction.

So, don’t start your paper like this:

Let MM be a complete Riemannian manifold, GG a compact Lie group and PMP \to M a principal GG-bundle.

Instead, try something more like this:

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.

But then quickly start explaining what progress you’ve made.

Use Setting and Context

In a short story, the reader is usually “located” as an observer to some scene of action, with a definite point of view — perhaps in a room somewhere, perhaps in some character’s mind, or whatever. The story should quickly and unobtrusively establish this context.

In a typical math paper, setting the scene is usually done in the introduction. This section explains the main results in more detail than the abstract, and put these results in their historical and mathematical context.

It’s very hard to appreciate a piece of mathematics without the necessary background. At the very simplest level, we need to understand all the words: mathematics bristles with technical terminology. So, the introduction to a math paper should set the scene as simply as possible, with a minimum of fancy vocabulary. 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. In this case, you must warn the reader that you’re doing so, and point the reader to the precise statement.

Readers often return over and over to the introduction for guidance as they struggle to understand a paper. So, ideally your introduction will not only set the stage, but serve as a road-map of the main concepts and results. Someone reading only the introduction should get a good idea of what your paper is about. Even if they don’t read it today, this may encourage them to return to it later.

Develop Your Characters

If a mathematics paper is secretly like a story, the “characters” are the mathematical entities involved. Some of these characters are more important than others; there are usually just a few heroes — and sometimes villains.

We can see the importance of developing characters by our tendency to singularize the plural. When we prove a theorem about a class of spaces X nX_n depending on some number nn, we often tell the reader to “fix” nn: that is, pick one, without saying which. Then we talk as if we were dealing with a particular space: a representative of the class under discussion. While this sort of move has been thoroughly analyzed by logicians, the art of good story-telling is also at work here. It is harder to keep in mind a class of entities than a particular representative of that class. Even authors of the crudest sort of politically engaged fiction, seeking to depict the “plight of the working class”, know enough to tell their story about a particular member, not the whole class all at once.

For your paper to be enjoyable, the main characters must be introduced in a way that marks them as special and highlights their already known properties: their “personality”. Don’t be afraid to say some things about them that the reader may already know. And when the hero arrives, there should be a little flourish of trumpets, like:

And now we come to a key player: the group of deck transformations.

Create Conflict and Tension

The “conflict” in a mathematics paper is usually the struggle to understand — often manifested in the struggle to prove something. As Piet Hein noted,

Problems worthy of attack
prove their worth by fighting back.

The most famous conjectures gain their interest from the way truths resist being known, forcing us into hard work and brand new insights. So if the results in your paper are harder to prove or less complete than you’d like, don’t feel too bad — played right, it can give your paper a touch of drama.

Alas, mathematicians are often too eager to play down the difficulties they faced. This not only makes math boring, it can make it harder to understand: a clever idea often seems unmotivated and mysterious unless one sees the problems it manages to overcome or circumvent. Of course, one should not give a detailed blow-by-blow account of every pitfall and wrong turn. And if every step in the final writeup is beautiful and well-motivated, there may be little scope for conflict and tension. But this is rare.

Find a Resolution

The conclusion of a math paper should set our feelings at rest by assuring us those problems that have been solved have indeed been solved, while reminding us of those that have not yet been solved.

All too often a math paper will end abruptly right after the main result has been proved. This is unpleasant, like lowering the curtain and turning on bright lights the instant after a movie reaches its climax. Are you really so eager to leave? Don’t be afraid to linger with the reader for a while and talk with them about a few topics that didn’t fit into the main flow of the argument. They will also enjoy hearing about open problems they could try to solve.


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.

This paper was originally intended to become a contribution to a book, which I recommend for many insights on the role of narrative in mathematics [DM].


[DM] A. Doxiadis and B. Mazur, eds., Circles Disturbed: the Interplay of Mathematics and Narrative, Princeton U. Press, Princeton, 2012.

[K] K. Kennedy, Short stories: 10 tips for novice creative writers.

Posted at March 28, 2024 10:21 PM UTC

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Re: Why Mathematics is Boring

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

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

Posted by: Mark Meckes on March 28, 2024 11:27 PM | Permalink | Reply to this

Re: Why Mathematics is Boring

Thanks! Yes, I realized that if I’m arguing against boring writing, people will read this with a very critical eye.

Posted by: John Baez on March 29, 2024 12:20 AM | Permalink | Reply to this

Re: Why Mathematics is Boring

That title sounds familiar!

Posted by: Tom Leinster on March 29, 2024 12:12 AM | Permalink | Reply to this

Re: Why Mathematics is Boring

Yes! Part of why this paper took so long to write is that the file was called boring.tex.

Posted by: John Baez on March 29, 2024 12:18 AM | Permalink | Reply to this

Re: Why Mathematics is Boring

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.

Posted by: Mike Shulman on March 29, 2024 3:23 AM | Permalink | Reply to this

Re: Why Mathematics is Boring

This is an excellent point. I’ve sometimes seen people argue against a story-telling approach to mathematical writing by saying that no one wants or needs to hear about all the dead ends. But that’s a straw-man argument; you don’t have to tell that story. There’s a whole genre of mathematical expository articles and talks with “You could have invented X” titles, and I think the philosophy behind that title is a good guide.

A useful analogy might be writing fictionalized accounts of historical events, in which all manner of things get simplified (or even greater liberties are taken) in order to tell a compelling story in limited time.

Posted by: Mark Meckes on March 29, 2024 3:06 PM | Permalink | Reply to this

Re: Why Mathematics is Boring

Thank, Mike! I will add a comment to that effect. I’d written

Of course, one should not give a detailed blow-by-blow account of every pitfall and wrong turn.

but this doesn’t clearly make the point that the story one is telling does not need to be historical at all: one gets to design it any way one wants.

When looking around for examples of interesting introductions, I checked out Andrew Wiles’ 1995 paper Modular elliptic curves and Fermat’s Last Theorem. His abstract is in Latin — always a winning move. He starts with some history of work on the problem. Then at the bottom of the second page he says:

Now we present our methods and results in more detail.

This intensely mathematical section goes on to page 8 and then he says:

The following is an account of the origins of this work and of the more specialized developments of the 1980’s that affected it. I began working on these problems in the late summer of 1986 immediately on learning of Ribet’s result. For several years I had been working on the Iwasawa conjecture for totally real fields and some applications of it. In the process, I had been using and developing results on \ell-adic representations associated to Hilbert modular forms. It was therefore natural for me to consider the problem of modularity from the point of view of \ell-adic representations. I began with the assumption that the reduction of a given ordinary \ell-adic representation was reducible and tried to prove under this hypothesis that the representation itself would have to be modular. I hoped rather naively that in this situation I could apply the techniques of Iwasawa theory.

And this very personal, yet also very mathematical, account goes on until the end of the introduction on page 11. So here is a case where an author thought it reasonable to present a detailed blow-by-blow account of what they actually did. I’d say this should be the exception, not the rule.

Posted by: John Baez on March 29, 2024 5:15 PM | Permalink | Reply to this

Re: Why Mathematics is Boring

Good mathematical exposition requires quite a bit of lying, in fact!

Posted by: Mariano Suarez Alvarez on April 2, 2024 1:54 AM | Permalink | Reply to this

Re: Why Mathematics is Boring

I’ve long felt that one of most mathematicians’ biggest problems with good exposition (as compared to physicists, computer scientists, etc) is not appreciating the value of useful lies.

Posted by: Mark Meckes on April 2, 2024 2:04 AM | Permalink | Reply to this

Re: Why Mathematics is Boring

I believe it’s okay to lie in math exposition (i.e., state results that aren’t true because they’re oversimplified) iff you say you are lying. You don’t want to stuff people’s heads with misinformation. Words like ‘roughly’, ‘essentially’, ‘oversimplifying’, etc. are ways to warn your readers not to add what you’re saying to their mental collection of theorems.

Posted by: John Baez on April 2, 2024 8:42 PM | Permalink | Reply to this

Re: Why Mathematics is Boring

Agreed! Lies are only useful when they are recognized as such.

Personally I’m fond of summarizing they hypotheses of a theorem with phrases like “in nice situations”. But some mathematicians seem allergic to even that level of simplification and insist on spelling out every detail to an audience that neither wants nor will remember it.

Posted by: Mark Meckes on April 3, 2024 1:39 PM | Permalink | Reply to this

Re: Why Mathematics is Boring

Mark wrote:

But some mathematicians seem allergic to even that level of simplification and insist on spelling out every detail to an audience that neither wants nor will remember it.

I can understand the impulse when someone is giving a seminar talk about a theorem they just proved: it’s their baby, and they are proud of it. But it’s much better to convey no more information than can easily be remembered. People will read the paper if they want details!

Posted by: John Baez on April 3, 2024 7:37 PM | Permalink | Reply to this

Re: Why Mathematics is Boring

Makes sense to have ChatGPT tailor papers to audidence.

Posted by: Tony Stark on March 29, 2024 2:04 PM | Permalink | Reply to this

Re: Why Mathematics is Boring


Posted by: John Baez on March 29, 2024 5:16 PM | Permalink | Reply to this

Re: Why Mathematics is Boring

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. :)

Posted by: Hein Hundal on March 29, 2024 2:27 PM | Permalink | Reply to this

Re: Why Mathematics is Boring

Great! Somewhat to my chagrin, my most cited paper is a review article on a somewhat quirky algebra called the octonions. I sometimes wonder how much this is due to the first paragraph of the introduction. I don’t usually go for such a humorous effect. But you can gauge how flashy you want your introduction to be.

Posted by: John Baez on March 29, 2024 5:20 PM | Permalink | Reply to this

Re: Why Mathematics is Boring

To be fair, it’s probably mostly because and nobody ever wrote an easily citable (let alone enjoyably readable) source on the octonions before, whereas nobody needs a reference to cite when they’re writing about the real or complex numbers. But surely that first paragraph doesn’t hurt.

Posted by: Mark Meckes on March 29, 2024 5:35 PM | Permalink | Reply to this

Re: Why Mathematics is Boring

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.

Posted by: Dr. Jason Polak on March 29, 2024 8:54 PM | Permalink | Reply to this

Re: Why Mathematics is Boring

I agree that good exposition is urgently needed in mathematics. And I don’t thinking spending a lot of time on it has hurt my career. On the contrary, whenever people invite me to give talks they tend to mention my expository series This Week’s Finds. You could even say I got a free year-long trip to Edinburgh based on this! So I think it pays to try to explain known math well.

Posted by: John Baez on March 30, 2024 12:09 AM | Permalink | Reply to this

Re: Why Mathematics is Boring

On the contrary, whenever people invite me to give talks they tend to mention my expository series This Week’s Finds.

Well, I think partially you reached a threshhold because you are well-known and more experienced. Junior researchers and lesser known individuals are not looked upon in the same way. Of course, you will not experience negative career impacts in the same way as many others.

Posted by: Dr. Jason Polak on March 30, 2024 12:13 PM | Permalink | Reply to this

Re: Why Mathematics is Boring

What you say may be true. I’m “well known” not despite the fact that I spent a lot of time explaining math and physics on the internet when I could have been writing papers, but because of it. But it’s possible that this is a hard road for people to follow now. I started when the internet was young, so it was easy to stand out. Now we have young mathematicians putting well-produced, entertaining yet serious videos on topics like sporadic finite simple groups, and online magazines like Quanta with slick explanations of things like Markov numbers. So the standards for math exposition are much higher.

Of course that’s good in many ways. If I had grown up in the 2000s I would have learned math much faster.

Posted by: John Baez on March 30, 2024 6:45 PM | Permalink | Reply to this

Re: Why Mathematics is Boring

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.)

Posted by: Sophie Morin on March 29, 2024 10:02 PM | Permalink | Reply to this

Re: Why Mathematics is Boring

Thanks! I’d never heard of the CARS model. Here’s a quick introduction for anyone interested:

The general model here reminds me most of papers in the social sciences and humanities, less of papers in the physical sciences, and still less of papers in math. For example, mathematicians often don’t try to justify their work by saying previous work was deficient in some way. We often take more of a ‘coral reef’ approach where we ask no more of each author than for them to contribute their own little bit to the growing structure. (But sometimes things get more exciting.)

Posted by: John Baez on March 30, 2024 12:23 AM | Permalink | Reply to this

Re: Why Mathematics is Boring

A coral reef! What a nice metaphor!

Hopefully we won’t die out en masse due to climate change too… 🤞

Posted by: Matteo Capucci on March 30, 2024 10:27 AM | Permalink | Reply to this

Re: Why Mathematics is Boring

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.

Posted by: John Baez on March 30, 2024 1:42 AM | Permalink | Reply to this

Re: Why Mathematics is Boring

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.

Posted by: Dave Cole on March 30, 2024 1:12 PM | Permalink | Reply to this

Re: Why Mathematics is Boring

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.

Posted by: Mozibur Ullah on April 1, 2024 7:37 AM | Permalink | Reply to this

Re: Why Mathematics is Boring

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

I think part of what’s happening here is that mathematicians’ brains have a tendency to condense such things into a not-commonly-divided unit that no longer has to be parsed as the conjunction of its pieces. I read that sentence and thought “what long string of adjectives?”—I literally had to force myself to re-read it to see the adjectives individually.

Assuming that these really are the appropriate conditions for one’s study, one is forced either to use them as they are, or to come up with new names for conjunctions of old conditions. This latter may seem friendlier, but as someone who’s never known as much commutative algebra as I should, I find myself more intimidated by the profusion of one-word terms that decompose, in some fashion I can never remember (and whose etymology can be at least as much historical as mathematical), into numerous but relatively concrete pieces. (A compromise that often gets adopted is abbreviations, which, of course, have their own drawbacks: LCHTVS is probably about as short as it gets, but not very friendly—and we run the risk of succumbing to the engineer’s disease of having an abbreviation where one letter stands for another abbreviation ….)

Posted by: L Spice on April 2, 2024 12:37 AM | Permalink | Reply to this

Re: Why Mathematics is Boring

I missed out an adjective. It really ought to be Locally convex Hausdorff real topological vector space. But I’ve come across the abbreviation LCHTVS -and its one of the uglier abbreviations out there - and it doesn’t mention real.

Now having the ‘convex’ qualifier around means our mathematical gadget needs to be a real vector space. Having the ‘locally’ qualifier means it must be topologised otherwise we can’t do locally anything. This does not mean the operations are cts. But a standing convention I have is that a topologised mathematical structure is by default a topological mathematical struct. This means all functions, operations and relations involved in the structure are required to be cts. Thus we have a real topological vector space.

(I do realise, btw, that topologised structures which are not topological are important. One important example are conveniant vector spaces - their addition is not cts in the c^Infinity topology. But my assumption I think is valid given the vast array of mathematical objects where this is the case. And it is also the natural assumption to make).

So locally convex is enough to deduce we have a LCTVS. Thus my preference is to simply call this LCvx. It’s shorter than LCTVS by one letter but also for legibility it only uses two capitals whereas the usual abbreviation uses five. Also, in its favour, is that it almost follows the usual namimg convention for proper names in English.

As for LCHTVS, I use instead HsLCvx or one could LCvxHs - the qualifier commutes with the proper name.

“… and whose etymology can be at least as much historical as mathematical…”

Mathematical history is mathematics. I often find that the motivation for thinking about certain ideas and structures are often motivated by its history than by the bare mathematics. Why should we care about diagram geometries? My guess is - and I’m no expert - is that one good reason is to come up with the analogue of Dynkin diagrams for finite simple groups.

Posted by: Mozibur Ullah on April 2, 2024 6:28 AM | Permalink | Reply to this

Re: Why Mathematics is Boring

People rarely keep saying ‘real’ or ‘complex’ vector space throughout their paper: they usually pick one, and tell the reader which one, right near the start of their paper. But of course there are some papers that need to use both, and then you’re stuck saying which one each time.

People talk a lot about ‘LCA’ groups, which are locally compact abelian Hausdorff groups. Some people, and Wikipedia, also call these locally compact abelian groups — leaving out the word ‘Hausdorff’ even though they are including that property!

I don’t really like acronyms, so if I wrote a paper that proved lots of theorems about locally convex Hausdorff topological vector spaces, I might say “henceforth all topological vector spaces are assumed to be locally convex and Hausdorff”, but remind people of that in statements of theorems. Or some trick like that.

Posted by: John Baez on April 2, 2024 8:10 PM | Permalink | Reply to this

Re: Why Mathematics is Boring

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.

Posted by: Mozibur Rahman Ullah on April 2, 2024 6:30 AM | Permalink | Reply to this

Re: Why Mathematics is Boring

I try to avoid saying any result is trivial, or obvious, or easy, or straightforward, or simple. There are a couple reasons:

1) If a fact is really trivial, you can either include the proof (because the proof is so short) or simply state the fact without proof (because either your readers can figure out the proof, or else they can’t — in which case telling them the proof is trivial won’t help them very much).

2) I make most of my mistakes when I claim some result is trivial and don’t prove it. The reason: the temptation to say it’s trivial is strongest when I feel it should be quick to prove, but I haven’t actually worked out a quick proof. For results that are really trivial, I’m not even tempted to say they’re trivial! I just state them as facts. Like, I would never say “It is trivial that 57/3 = 19”.

In short, the temptation to say a result is trivial, or obvious, or easy, or straightforward, or simple, often arises from some sort of inner mental conflict about whether to provide a proof or not.

Similarly, to say a result is well-known is fine if you include a precise reference, but otherwise it signals that you are either too lazy to look for a reference, or you looked and couldn’t actually find one.

Posted by: John Baez on April 2, 2024 8:18 PM | Permalink | Reply to this

Re: Why Mathematics is Boring

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.

Posted by: a different heart of the matter on April 3, 2024 12:10 AM | Permalink | Reply to this

Re: Why Mathematics is Boring

More discussion over on HN.

Posted by: Simon Burton on April 3, 2024 4:26 PM | Permalink | Reply to this

Re: Why Mathematics is Boring

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

Posted by: Ian on April 4, 2024 7:30 PM | Permalink | Reply to this

Re: Why Mathematics is Boring

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.

Posted by: Enrique Treviņo on April 6, 2024 3:03 PM | Permalink | Reply to this

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