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<< Counting Points on Elliptic Curves (Part 3) | Main

March 28, 2024

Why Mathematics is Boring

Posted by John Baez

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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!

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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 P-MP \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.

Conclusion

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].

References

[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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21 Comments & 0 Trackbacks

Re: Why Mathematics is Boring

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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

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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

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That title sounds familiar!

Posted by: Tom Leinster on March 29, 2024 12:12 AM | Permalink |
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Re: Why Mathematics is Boring

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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

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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

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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

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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

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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

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No.

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

Re: Why Mathematics is Boring

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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

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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

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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

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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 |
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Re: Why Mathematics is Boring

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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

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    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 |
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Re: Why Mathematics is Boring

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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

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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 |
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Re: Why Mathematics is Boring

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

  * John Swales, "Create a Research Space" (CARS) model of research
    introductions.

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

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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 |
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Re: Why Mathematics is Boring

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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

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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

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