What math stories to tell and not to tell?

Storytelling can be surprisingly powerful. When a story is skillfully told, you get an almost magical feeling of being a part of it, making you care deeply about protagonists. Even if under ordinary circumstances you have zero empathy for the Civil War era outlaws or emperor penguins of Antarctica, you suddenly may find yourself engrossed with their fortune. This is a difficult skill to master, but the effects are visible even when used in earnest by the beginners.

Recently I started thinking about the kind of stories mathematicians should be telling. This was triggered by Angela Gibney‘s kind invitation to contribute an article on math writing to the Early Career Collection in the Notices of the AMS. So I looked at a few older articles and found them just wonderful. I am not the target audience for some of them, but I just kept reading them all one after another until I exhausted the whole collection.

My general advice — read the collection! Read a few pieces by some famous people or some people you know. If you like them, keep on reading. As I wrote in this blog post, you rarely get an insight into mathematician’s thinking unless they happen to write an autobiography or gave an interview. While this is more of a “how to” genre, most pieces are written in the first person narrative and do tell some interesting stories or have some curious points of view.

It is possible I am the last person to find out about the collection. I am not a member of the AMS, I don’t read the Notices, and it’s been a long time since anyone considered me “early career”. I found a few articles a little self-centered (but who am I to judge), and I would quibble with some advice (see below). But even those articles I found compelling and thought-provoking.

Having read the collection, I decided to write about mathematical storytelling. This is not something that comes naturally to most people in the field. Math stories (as opposed to stories about mathematicians) tend to be rather dry and unexciting, especially in the early years of studying. I will blog my own article some other time, but for now let me address the question in the title.

Stories to tell

With a few notable exceptions, just about all stories are worth telling. Whether in your autobiography or in your personal blog, as long as they are interesting to somebody — it’s all good. Given the lack of good stories, or any math stories really, it’s a good bet somebody will find your stories interesting. Let me expound on that.

Basically, anything personal works. To give examples from the collection, see e.g. stories by Mark Andrea de Cataldo, Alicia Prieto-Langarica, Terry Tao and John Urschel. Most autobiographies are written in this style, but a short blog post is also great. Overcoming an embarrassment caused by such public disclosure can be difficult, which makes it even more valuable to the readers.

Anything historical works, from full length monographs on history of math to short point of view pieces. Niche and off the beaten path stories are especially valuable. I personally like the classical History of Mathematical Notations by Florian Cajori, and Combinatorics: Ancient & Modern, a nice collection edited by Robin Wilson and John Watkins, with a several articles authored by names you will recognize. Note that an oral history can be also very valuable, see the kind of stories discussed by László Lovász and Endre Szemerédi mentioned in this blog post and Dynkin’s interviews I discussed here.

Anything juicy works. I mean, if you have a story of some famous mathematician doing something unusual (good or bad, or just plain weird), that attracts attention. This was the style of Steven Krantz’s two Math Apocryphia books, with many revealing and embarrassing anecdotes giving a sense of the bygone era.

Anything inspirational works. A beautiful example of this style is Francis Su’s Farewell Address as MAA President and part of his moving follow up book (the book has other interesting material as well). From the collection, let me single out Finding Your Reward by Skip Garibaldi which also aims to inspire. Yet another example is Bill Thurston‘s must read MO answer “What’s a mathematician to do?“

Any off the beaten path math style is great. Think of “The Strong Law of Small Numbers” by Richard Guy, or many conjectures Terry Tao discusses in his blog. Think of “Missed opportunities” by Freeman Dyson, “Tilings of space by knotted tiles” by Colin Adams, or “One sentence proof… ” by Don Zagier (see also a short discussion here) — these are all remarkable and memorable pieces of writing that don’t conform to the usual peer review paradigm.

Finally, anything philosophical or metamathematical finds an audience. I am thinking of “Is it plausible?” by Barry Mazur, “Theorems for a Price” by Doron Zeilberger, “You and Your Research” by Richard Hamming, “Mathematics as Metaphor” by Yuri Manin, or even “Prime Numbers and the Search for Extraterrestrial Intelligence” by Carl Pomerance. We are all in search of some kind of answers, I suppose, so reading others thinking aloud about these deep questions always helps.

Practice makes perfect

Before I move to the other side, here is a simple advice on how to write a good story. Write as much as possible! There is no way around this. Absolutely no substitute, really. I’ve given this advice plenty of times, and so have everyone else. Let me conclude by this quote by Don Knuth which is a bit similar to Robert Lazarsfeld‘s advice. It makes my point much better and with with more authority that I can ever provide:

Of equal importance to solving a problem is the communication of that solution to others. The best way to improve your writing skills is to practice, practice, practice.

Seize every opportunity to write mini-essays about the theoretical work you are doing. Compose a blog for your friends, or even for yourself. When you write programs, write literate programs.

One of the best strategies to follow while doing PhD research is to prepare weekly reports of exactly what you are doing. What questions did you pursue that week? What positive answers did you get? What negative answers did you get? What are the major stumbling blocks that seem to be present at the moment? What related work are you reading?

Donald Knuth – On Writing up Research (posted by Omer Reingold), Theory Dish, Feb 26, 2018

Don’t be a journalist

In this interesting article in the same collection, Jordan Ellenberg writes:

Why don’t journalists talk about math as it really is? Because they don’t know how it really is. We do. And if we want the public discourse about math to be richer, broader, and deeper, we need to tell our own stories.

He goes on to suggest that one should start writing a blog and then pitch some articles to real newspapers and news magazines. He gives his own bio as one example (among others) of pitching and publishing in mainstream publications such as Slate and the New York Times. Obviously, I agree with the first (blog) part (duh!), but I am rather negative on the second part. I know, I know, this sounds discouraging, but hear me out.

First, what Jordan is not telling you is how hard he had to work on his craft before getting to the point of being acceptable to the general audience. This started with him getting Summa Cum Laude A.B. degree from Harvard in both Math and English (if I recall correctly), and then publishing a well-received novel, all before starting his regular Slate column. Very few math people have this kind of background on which they can build popular appeal.

Second, this takes away jobs from real journalists. Like every highly competitive intellectual profession, journalism requires years of study and practice. It has its own principles and traditions, graduate schools, etc. Call it a chutzpah or a Dunning–Kruger effect, but just because you are excellent in harmonic analysis doesn’t mean you can do even a mediocre job as a writer. Again — some people can do both, but most cannot. If anything, I suspect a negative correlation between math and writing skills.

Here is another way to think about this. Most people do realize that they don’t need to email their pretty iPhone pictures of a Machu Picchu sunrise to be published by the National Geographic. Or that their cobbler family recipe maybe not exactly be what Gourmet Magazine is looking for. Why would you think that writing is much easier then?

Third, this cheapens our profession to some degree. You really don’t need a Ph.D. in algebraic number theory and two perfect scores at the IMO to write about Powerball or baseball. You need a M.S. in statistics and really good writing skills. There are plenty of media sites which do that now, such as 538. There is even the whole DDJ specialization with many practitioners and a handful of Pulitzer prizes. Using quantitative methods is now mainstream, so what exactly are you bringing to the table?

Fourth, it helps to be honest. Jordan writes: “Editors like an angle. If there’s a math angle to a story in the news, pitch it! As someone with a degree in math, you have something to offer that most writers don’t.” This is true in the rare instances when, say, a Fields medal in your area is awarded, or something like that. But if it’s in an area far away from yours, then, uhm, you got nothing over many thousands of other people.

Now, please don’t take this as “don’t comment on current affairs” advice. No, no — please do! Comment away on your blog or on your podcast. Just don’t take jobs away from journalists. Help them instead! Write them emails, correct their mistakes. Let them interview you as an “expert”, whatever. Part of the reason the math related articles are so poor is because of mathematicians’ apathy and frequent disdain to the media, not because we don’t write newspaper articles — it’s really not our job.

Let me conclude with an anecdote about me reaching out to a newspaper. Once upon a time, long ago, flights used to distribute real newspapers to the passengers. I was sitting in the back and got a Wall Street Journal which I read out of boredom during takeoff. There was an article discussing the EU expansion and the fact that by some EU rules, the headquarters need a translator from every language to every other language. The article predicted dark days ahead, since it’s basically impossible to find people who can translate some smaller languages, such as from Maltese to Lithuanian. The article provided a helpful graph showing the number of translators needed as a function of the number of countries and claimed the exponential growth.

I was not amused, cut out the article, and emailed the author upon arrival, saying that with all my authority as an assistant professor at MIT, I promise that n(n-1) grows polynomially, not exponentially. I got back a surprisingly apologetic reply. The author confessed he was a math major in college, but was using the word without thinking. I don’t know if WSJ ever published a correction, but I bet the author will not be using this word so casually anymore, and if he ever advanced to the editorial position will propagate this knowledge to others. So there — that’s my personal contribution to improving public discourse…

Don’t be an apologist

In another beautifully written article in the Early Career collection, Izzet Coskun gives “advice on how to communicate mathematics quickly in informal settings”. He writes:

Whether before a promotion committee, at a party where one might meet future politicians or future parents of future colleagues, in the elevator on the way up to tea, or in the dean’s office at a job interview, we often have the opportunity to explain our work to a general audience. The time we have is usually short [..] Our audience will not be familiar with our terminology. Communicating mathematics in such settings is challenging.

He then gives a lot of very useful practical advice on how to prepare to such “math under a minute” conversation, how to be engaging, accessible, etc. It’s an all around good advice. However, I disagree. Here is my simple advice: Don’t Do It! If it’s a dean and this is a job interview, feel free to use any math jargon you want — it’s not your fault your field is technical, and the dean of sciences is used to it anyway. Otherwise, just say NO.

It’s true that sometimes your audience is friendly and is sincere in their interest in your work. In that case no matter what you say will disappoint them. There is a really good chance they can’t understand a word of what you say. They just think they can, and you are about to disillusion them.

But more often than not, the audience is actually not friendly, as was the case of a party Izzet described in his article. Many people harbor either a low regard or an outright resentment towards math stemming from their school years or some kind of “life experience”. These folks simply want to reinforce their views, and no matter what you say that will be taken as “you see, math is both hard, boring and useless”.

One should not confuse the unfriendlies with stupid or uneducated people. On the contrary, a lot of educated people think this way. A prime example is Amy Wax with her inimitable quote:

If we got rid of ninety percent of the math Ph.D. programs, would we really be worse off in any material respect?  I think that’s a serious question.

I discussed this quote at length in this blog post. There, I tried to answer her question. But after a few back-and-force emails (which I didn’t make public), it became clear that she is completely uninterested in the actual learning of what math is and what it does. She just wants to have her own answer validated by some area practitioners. Oh, well…

So here is the real reason why I think answering such people is pointless. No matter what you say, you come across as an apologist for the field. If people really want to understand what math is for, there are plenty of sources. In fact, have several bookshelves with extremely well written book-length answers. But it’s not your job to educate them! Worse, it is completely unreasonable to expect you to answer in “under one minute”.

Think about reactions of people when they meet other professionals. Someone says “I develop new DNA based cancer treatments” or “I work on improving VLSI architecture”, or “I device new option pricing strategies”. Is there a follow up request to explain it in “under one minute”? Not really. Let me give you a multiple choice. Is that because people think that:

a) these professions are boring compared to math and they would rather hear about the latter?

b) they know exactly what these professionals do, but math is so darn mysterious?

c) they know these professions are technical and hard to understand, but even children can understand math, so how hard can that be?

d) these professions are clearly useful, but what do math people do — solve quadratic equations all day?

If you answered a) or b) you have more faith in humanity than I do. If you answered c) you never spoke to anyone about math at a party. So d) is the only acceptable answer, even if it’s an exaggeration. Some people (mostly under 7) think that I “add numbers all day”, some people (mostly in social sciences) think that I “take derivatives all day”, etc., you get the point. My advice — don’t correct them. This makes them unhappy. Doesn’t matter if they are 7 or 77 — when you correct them the unhappiness is real and visible…

So here is a summary of how I deal with such questions. If people ask what I do, I answer “I do math research and I teach“. If they ask what kind of research I say “advanced math“. If they ask for details I tell them “it’s complicated“. If they ask why, I tell them “because it takes many years of study to even understand the math lingo, so if I tell you what I do this sounds like I am speaking a foreign language“.

If they ask what are the applications of my research (and they always do), I tell them “teaching graduate classes“. If they ask for “practical” applications, whatever that means, I tell them “this puts money into my Wells Fargo account“. At this point they move on exhausted by the questions. On the one hand I didn’t lie except in the last answer. On the other — nobody cares if I even have a WF account (I don’t, but it’s none of their business either).

One can ask — why do I care so much? What’s so special about my work that I am so apprehensive? In truth, nothing really. There are other aspects of my identity I also find difficult discussing in public. The most relevant is “What is Combinatorics?” which for some reason is asked over and over as if there is a good answer (see this blog post for my own answer and this Wikipedia article I wrote). When I hear people explaining what it is, half the time it sounds like they are apologizing for something they didn’t do…

There are other questions relevant to my complex identity that I am completely uninterested in discussing. Like “What do you think of the Russian President?” or “Who is a Jew?“, or “Are you a Zionist?” It’s not that my answers are somehow novel, interesting or controversial (they are not). It’s more like I am afraid to hear responses from the people who asked me these questions. More often than not I find their answers unfortunate or plain offensive, and I would rather not know that.

Let me conclude on a positive note, by telling a party story of my own. Once, during hors d’oeuvres (remember those?), one lady, a well known LA lawyer, walked to me and said: “I hear you are a math professor at UCLA. This is so fascinating! Can you tell me what you do? Just WOW me!” I politely declined along the lines above. She insisted: “There has to be something that I can understand!” I relented: “Ok, there is one theorem I can tell you. In fact, this result landed me a tenure.” She was all ears.

I continued: “Do you know what’s a square-root-of-two?” She nodded. “Well, I proved that this number can never be a ratio of two integers, for example it’s not equal to 17/12 or anything like that.” “Oh, shut-the-F-up!” she exclaimed. “Are you serious? You can prove that?” — she was clearly suspicious. “Yes, I can“, I confirmed vigorously, “in fact, two Russian newspapers even printed headlines about that back a few years ago. We love math over there, you know.”

“But of course!“, she said, “American media never writes about math. It’s such a shame! That’s why I never heard of your work. My son is much too young for this, but I must tell my nieces — they love science!” I nodded approvingly. She drifted away very happy, holding a small plate of meat stuffed potato croquettes, enriched with this newly acquired knowledge. I do hope her nieces liked that theorem — it is cool indeed. And the proof is so super neat…

Mathematician’s guide to holidays

Holiday season offers endless opportunities to celebrate, relax, rest, reflect and meditate.  Whether you are enjoying a white Christmas or a palm tree Chanukkah, a mathematician in you might wonder if there is more to the story, a rigorous food for thought, if you will.  So here is a brief guide to the holidays for the mathematically inclined.

1)  Christmas tree lectures

I have my own Christmas tree tradition.  Instead of getting one, I watch new Don Knuth‘s “Christmas tree lecture“.  Here is the most recent one.  But if you have time and enjoy binge-watching here is the archive of past lectures (click on “Computer musings” and select December dates).  If you are one of my Math 206 students, compare how Knuth computed the number of spanning trees in a hypercube (in a 2009 lecture) with the way Bernardi did in his elegant paper.

2)  Algorithmic version of Fermat’s Christmas theorem

Apparently, Fermat’s theorem on sums of two squares first appeared in Fermat’s long letter to Mersenne, written on Christmas Day (December 25, 1640).  For background, see Catalan and French language Wikipedia articles.  Zagier’s “one-sentence proof” is well known and available here.  Long assumed to be mysterious, it was nicely explained by Elsholtz.  Still mysteriously, a related proof also appears in a much earlier paper (in French), by a Russian-American mathematician J. Uspensky (ht. Ustinov).  Can somebody explain to me what’s in that paper?

Interestingly, there is a nice polynomial time algorithm to write a prime  p=1 mod 4  as a sum of two squares, but I could not find a clean version on the web.  If you are curious, start with Cornacchia’s algorithm for more general quadratic Diophantine equations, and read its various proofs (advanced, elementary, short, textbook, in French).  Then figure out why Fermat’s special case can be done in (probabilistic) polynomial time.

3)  Dreidel game analysis

The dreidel is a well known Chanukkah game with simple rules.  Less known is the mathematics behind it.  Start with this paper explaining that it’s unfair, and continue to this paper explaining how to fix it (on average).  Then proceed to this “squared nuts” conjecture by Zeilberger on the expected length of the game (I have a really good joke here which I will suppress).  This conjecture was eventually resolved in this interesting paper, definitely worth $25 promised by Zeilberger.

Now, if you are underwhelmed with the dreidel game, try to prove the festive Star of David Theorem.  When you are done, enjoy this ingenious proof, which is definitely “from the book”.

4)  Santa Claus vs beautiful mathematics

Most readers of this blog are aware of existence of beautiful mathematics.  I can only speculate that a clear majority of them would probably deny the existence of Santa Claus.  However, there are millions of (mostly, very young) people who believe the exact opposite on both counts.  Having grown up in the land of Ded Moroz, we have little to say on the great Santa debate, but we believe it’s worth carefully examining Santa proponent’s views.  Could it be that their arguments can be helpful in our constant struggle to spread the gospel of beautiful mathematics?

We recommend reading “Yes, Virginia, there is Santa Claus“ column (fully available here), which was originally published by the New York Sun in 1897.  In fact, read it twice, three times, even four times.  I am reluctant to quote from it because it’s short and deserves to be read in full.  But note this passage:  “The most real things in the world are those that neither children nor men can see.”  The new Jewish editor of the Sun reports that the rabbis he consulted think this is “a joyous articulation of faith”.  Maybe.  But to me this evokes some beautiful advanced mathematics.

You see, when mathematicians try to explain that mathematics is beautiful, they tend to give simple visually appealing examples (like here).  But I suggest closing your eyes and imagining beautiful mathematical objects, such as the 600-cell, Poincaré homolgy sphere, Lie group E₈, Monster group, or many other less known higher dimensional constructions such as the associahedron, the Birkhoff polytope, Walz’s flexible cross-polyhedron, etc.  Certainly all of these can be seen by “neither children nor men”.  Yet we can prove that they “are real”.  We can then spend years studying and generalizing them.  This knowledge alone can bring joy to every holiday season…

HAPPY HOLIDAYS EVERYONE!   С НОВЫМ ГОДОМ!

On triple crowns in mathematics and AMS badges

As some of you figured out from the previous post, my recent paper (joint with Martin Kassabov) was accepted to the Annals of Mathematics.  This being one of my childhood dreams (well, a version of it), I was elated for a few days.  Then I thought – normal children don’t dream about this kind of stuff.  In fact, we as a mathematical community have only community awards (as in prizes, medals, etc.) and have very few “personal achievement” benchmarks.  But, of course, they are crucial for the “follow your dreams” approach to life (popularized famously in the Last Lecture).  How can we make it work in mathematics?

I propose we invent some new “badges/statistics” which can be “awarded” by AMS automatically, based on the list of publications, and noted in the MathSciNet Author’s Profile.  The awardees can then proudly mention them on the department websites, they can be included in Wikipedia entries of these mathematicians, etc.   Such statistics are crucial everywhere in sports, and most are individual achievements.  Some were even invented to showcase a particular athlete.   So I thought – we can also do this.  Here is my list of proposed awards. Ok, it’s not very serious…  Enjoy!

Triple Crown in Mathematics

A paper in each of Annals of Mathematics, Inventiones, and Journal of AMS.  What, you are saying that “triple crown” is about horse racing?  Not true.  There are triple crowns in everything, from bridge to golf, from hiking to motor racing.  Let’s add this one to the list.

Other Journal awards

Some (hopefully) amusing variations on the Tripe Crown.  They are all meant to be great achievements, something to brag about.

Marathon – 300 papers

Ultramarathon – 900 papers

Iron Man – 5 triple crown awards

Big Ten – 10 papers in journals where “University” is part of the title

Americana – 5 papers in journals whose title may only include US cities (e.g. Houston), states (e.g. Illinois, Michigan, New York), or other parts of American geography (such as Rocky Mountains, Pacific Ocean)

Foreign lands – 5 papers in journals named after non-US cities (e.g. Bordeaux, Glasgow, Monte Carlo, Moscow), and five papers in journals named after foreign countries.

Around the world – 5 papers in journals whose titles have different continents (Antarctica Journal of Mathematics does not count, but Australasian Journal of Combinatorics can count for either continent).

What’s in a word – 5 papers in single word journals: (e.g. Astérisque, Complexity, Configurations, Constraints, Entropy, Integers, Nonlinearity, Order, Positivity, Symmetry).

Decathlon – papers in 10 different journals beginning with “Journal of”.

Annals track – papers in 5 different journals beginning with “Annals of”.

I-heart-mathematicians – 5 papers in journals with names of mathematicians (e.g. Bernoulli, Fourier, Lie, Fibonacci, Ramanujan)

Publication badges

Now, imagine AMS awarded badges the same way MathOverflow does, i.e. in bulk and for both minor and major contributions.  People would just collect them in large numbers, and perhaps spark controversies.  But what would they look like?  Here is my take:

enthusiast (bronze) – published at least 1 paper a year, for 10 years (can be awarded every year when applicable)

fanatic (silver) – published at least 10 papers a year, for 20 years

obsessed (gold) – published at least 20 papers a year, for 30 years

nice paper (bronze) – paper has at least 2 citations

good paper (silver) – paper has at least 20 citations

great paper (gold) – paper has at least 200 citations

famous paper (platinum) – paper has at least 2000 citations

necromancer (silver) – cited a paper which has not been cited for 25 years

asleep at the wheel (silver) – published an erratum to own paper 10 years later

destroyer (silver) – disproved somebody’s published result by an explicit counterexample

peer pressure (silver) – retracted own paper, purchased and burned all copies, sent cease and desist letters to all websites which illegally host it

scholar (bronze) – at least one citation

supporter (bronze) – cited at least one paper

writer (bronze) – first paper

reviewer (bronze) – first MathSciNet review

self-learner (bronze) – solved own open problem in a later paper

self-citer (bronze) – first citation of own paper

self-fan (silver) – cited 5 own papers at least 5 times each

narcissist (gold) – cited 15 own papers at least 15 times each

enlightened rookie (silver) – first paper was cited at least 20 times

dry spell (bronze) – no papers for the past 3 years, but over 100 citations to older papers over the same period

remission (silver) – first published paper after a dry spell

soliloquy (bronze) – no citation other than self-citations for the past 5 years

drum shape whisperer (silver) – published two new objects with exactly same eigenvalues

neo-copernicus (silver) – found a coordinate system to die for

gaussian ingenuity (gold) – found eight proofs of the same law or theorem

fermatist (silver) – published paper has a proof sketched on the margins

pythagorist (gold) – penned an unpublished and publicly unavailable preprint with over 1000 citations

homologist (platinum) – has a (co)homology named after

dualist (platinum) – has a reciprocity or duality named after

ghost-writer (silver) – published with a person who has been dead for 10 years

prince of nerdom (silver) – wrote a paper joint with a computer

king of nerdom (gold) – had a computer write a joint paper

sequentialist (gold) – authored a sequel of five papers with the same title

prepositionist (gold) – ten papers which begin with a preposition “on”, “about”, “toward”, or “regarding” (prepositions at the end of the title are not counted, but sneered at).

luddite (bronze) – paper originally written NOT in TeX or LaTeX.

theorist (silver) – the implied constant in O(.) notation in the main result in greater than 10⁸⁰.

conditionalist (silver) – main result is a conditional some known conjecture (not awarded in Crypto and Theory CS until the hierarchy of complexity classes is established)

ackermannist (gold) – main result used a function which grows greater than any finite tower of 2’s.

What about you?  Do you have any suggestions? 🙂