Michael Boshernitzan, a math professor at my graduate school alma mater, Rice University, passed away last week. I took one class with him, but he had a larger impact in my life as my spouse Jon Chaika’s advisor. They had a wonderful advisor-student relationship, and I know Jon especially appreciated Michael’s keen eye for interesting math questions and limitless enthusiasm for math. (It feels odd for me to refer to my husband as Chaika instead of Jon. But it also feels odd to refer to one person in a post by first name and one by last name or to refer to both by first name. With no non-odd options available, I will refer to them as Michael and Jon.)

After hearing the sad news, I found myself browsing his apers on the arXiv and decided to click on “The densest sequence in the unit circle,” which he and Jon cowrote a decade ago. The paper is only two pages long, and it’s kind of fun.

The title of their paper refers to the unit circle, which is usually defined to be the circle with radius 1, but the unit circle they are actually using is the circle with a circumference of 1 (hence a radius of 1/π). The distinction doesn’t really matter—they could multiply things by π as needed to ship their work over to the standard unit circle—but this fact can help us understand part of how they did their work. (Usually when mathematicians refer to a circle, we are referring to a hollow circle, not a filled-in circle, which we would call a disc. Unfortunately, we are often sloppy about this usage, saying things like “the area of a circle is πr^{2}.” The area of a circle is 0 because a circle is 1-dimensional, not 2-dimensional. The *length* of a circle is 2πr, and the area of a *disc* is πr^{2}.)

One way to think of a circle is as an interval with its two ends glued together, in this case an interval with length 1. Doing arithmetic on this interval involves thinking about the “fractional part” of the numbers involved. The fractional part of a real number is what you get when you chop off anything to the left of the decimal point. The fractional part of 1.5 is 0.5, the fractional part of π is .14159…, the fractional part of 5 is 0. (As the π example shows, a number’s fractional part isn’t necessarily a fraction.) In the paper, Michael and Jon denote the fractional part of a number *x* by *x (mod 1)*.

Working with the fractional parts of numbers makes it easy to see the relationship between the number line and the circle and understand a big part of the paper: answering a question about points on the circle by doing arithmetic on the real number line. Dealing with fractional parts only, we can think about a number line being a bunch of intervals chopped up and stacked on top of each other, or better yet, as being wrapped around a circle infinitely many times.

Michael and Jon were looking for the densest way to fill up a circle with an infinite sequence of points. The only problem is it’s not obvious how to measure the density of a set of zero-dimensional points in a one-dimensional line. Part of the work in the paper, in fact, was to figure out a sensible ways to measure the property they were after.

The traditional definition of density for sets within the number line is that a set is dense if every small interval contains points in the set. Rational numbers are dense because any interval, no matter how small its length, contains at least one rational number. This notion of density is binary: a set is either dense or it is not. Many, many sequences of numbers in the circle (viewed as fractional parts of real numbers) are dense in this sense. For example, a sequence obtained by adding the same irrational number to the previous term over and over (and taking the fractional part of the answer) is dense in the circle. Eventually, some term of the sequence will hit any tiny interval in the circle. Adding a rational number over and over, on the other hand, is not dense because eventually you will be repeating the same number. For example, the sequence 0, 1/2, 1/2, 3/4, 0, 1/4, 1/2, 3/4,… only hits the same four numbers.

Many sequences are dense in this sense, but is there a way to quantify how quickly and effectively a sequence fills in the circle? Those are the notions Michael and Jon zeroed in on. They came up with two measures and a sequence that optimizes both of them to a certain extent.

The first measure is labeled D_{n} in the paper and measures the farthest away any point of the circle can be from the nearest point in the first *n* terms of the sequence. An example helps: Let’s pick a simple sequence, the sequence where the nth term is the number n/3 (mod 1). Starting at 0, this sequence goes 0, 1/3, 2/3, 0, 1/3, 2/3, and so on, just marching along the circle taking steps with length 1/3. After n=3, the density D_{n} stabilizes. The points 1/6, 1/2, and 5/6 are all halfway in between two of the points of the sequence, and they have distance 1/6 from the nearest point. Any other point on the circle is closer to a sequence point. So D_{n} is 1/6 for any *n* greater than or equal to 3. Calculating D_{n} for a more complicated sequence than 0, 1/3, 2/3 is of course going to be more challenging, but the big idea is that this measure keeps track of how badly the sequence misses hitting every point in the unit interval at the nth step for each *n*.

The other measure they used is called d_{n }(if you’re reading this post out loud, I’d recommend shouting D_{n }and whispering d_{n} to help keep them straight in your mind), and it keeps track of how close the closest points of the sequence are to each other. In the case of 0, 1/3, 2/3, the closest points have distance 1/3 from each other, so d_{n} for that sequence is 1/3. This notion might be better thought of as sprawl than density: A sequence with large values of d_{n} is a sequence where each new term is pretty far from the previous terms.

The goal of the paper is to find a sequence that does a good job with both D_{n }and d_{n}. That means small values of D_{n}—the sequence is getting close to all the points in the circle—and large values of d_{n}—each new term of the sequence is pretty far from all the previous terms.

The winning sequence Michael and Jon found is actually not too hard to understand. It’s log_{2}(2k−1) (mod 1) with k=1, 2, 3, and so on. The function log_{2}(x) is the base 2 logarithm of *x*, the number to which 2 must be raised to get *x*; for example, 2^{3}=8, so log_{2}(8)=3. Plugging in a few numbers, you can see that the sequence from the paper is the sequence of the fractional parts of the base 2 logarithms of odd whole numbers. That’s a mouthful, but it’s not hard to start plugging in numbers on an online calculator to get a feel for the beginning of the sequence. For small values of *k*, the fractional part of the base 2 logarithm of *k* moves around a lot. For large values of *k*, the sequence terms eventually get close together. (Try plugging in some 5- or 6-digit numbers for *k* and see how little the sequence moves from one term to the next.)

In their paper, Michael and Jon showed that the sequence of base 2 logs of odd numbers does a good job of both getting close to all the points in the circle and keeping the minimum distance between points of the sequence on the large side of what is possible. In particular, they show that competitor sequences may sometimes beat their sequence, but their sequence will win infinitely often. In fact, as *n* increases, their sequence gets winning-er and winning-er over all competitors.

This is not an earth-shattering paper, but I decided to dig into it because it was bite-sized and accessible to me, unlike some of his deeper work. It also fit the way I perceived Michael as a mathematician, though of course my perception was from a distance; I never worked with him. He liked asking and answering questions that were perfectly natural but no one had thought to ask before. While it took some elbow grease to set up in this blog post, the basic premise of the question would probably be straightforward to a math graduate student who had taken an introductory analysis class. Michael also loved clever, efficient arguments. After the basic terms are defined in this paper, the proofs of the two theorems only takes about half a page and doesn’t require heavy mathematical machinery so much as astute observations about sequences and intervals. Math doesn’t have to be earth-shattering to be worth thinking about, and I’m glad I spent some time thinking about this problem.

]]>Flyer (prints 2 per page)

Poster (prints 1 per page)

Lots of people in Utah are looking for a family-friendly, free activity that engages the body, mind, and spirit and lets them connect with others and be part of something bigger than themselves. Tell them about Sacred Harp singing!

]]>My neighborhood isn’t exactly sleepy, but the traffic isn’t too bad. Sometimes I can even cross intersections diagonally while I take a walk. On one recent walk, I started wondering how much distance I save myself with my little hypotenuse trick, so when I got home I measured our intersection. Blocks in my neighborhood are approximately 400 feet long, and streets are approximately 40 feet wide.

Based on those numbers, I used the Pythagorean theorem to figure out that I would walk about 460 feet if I walked along a block and cross an intersection on the diagonal or 480 feet if I crossed both streets separately, a savings of about 5% of the distance. It’s not terrible, but it’s not a lot either. Stopping to pet just one cat would probably wipe out any time savings on an average walk. (Worth it!)

In contrast, behold Chicago! In May, I stayed just off of Milwaukee Avenue during a visit to Chicago. Milwaukee runs from the west part of downtown Chicago almost all the way up to the Wisconsin border. It changes its precise compass bearing a few times during the trip, but it starts out at almost exactly a 45 degree angle, the perfect hypotenuse to the rest of Chicago’s street grid.

The event I was attending was northwest of my hotel, and I almost wept at the beautiful efficiency as I walked along Milwaukee. The unconventional six-way and mid-block intersections are a small price to pay for the pleasure of walking along a glorious hypotenuse. The city’s most-used bike lane is along Milwaukee. A cyclist can ride from their Logan Square abode through the chocolatey air of River West and to downtown in only 70% of the time it would take on the rectilinear streets of the grid.

Milwaukee has been an important road for Chicago since before white settlers established the city we now know by that name. It was used by Native Americans long before their land was taken by the American government, as this article about the avenue describes.

Milwaukee is not the only hypotenuse in Chicago. The city is lousy with them. Grand, Clybourn, Lincoln, Ogden, and several other streets cut jaunty angles across the city, disrupting the tidiness of the grid and saving pedestrians and cyclists a few steps or twirls of the pedals. The next time you’re in Chicago, after you’ve enjoyed the fantastic food, striking architecture, and beautiful lake, a long walk might help you appreciate the city’s hypotenuses.

]]>I transcribe every episode of My Favorite Theorem, the podcast I cohost with Kevin Knudson. It’s important for accessibility (and yes, I do judge you if you publish a podcast with no transcript, especially if you have a decent budget), but I’ve also found that a lot of hearing people prefer to read rather than listen to our podcast. Selfishly, having transcripts makes it possible to search text more easily in case I want to refer to something later.

A few months ago I started using otter.ai to help with my transcriptions. (This is not an Otter ad. It’s fine, but I think there are plenty of other companies that do similar work.) I upload the audio of the podcast, let the algorithm create a first draft of the transcript, and then listen to the podcast again and clean it up. Otter does a pretty decent job, but of course it is not perfect. It’s not too bad at everyday words, but its accuracy plummets on more technical terms or words from languages other than English, often to hilarious effect. In our episode with Jim Propp, it rendered “combinatorics” as “common at Oryx,” “enumeration of tilings” as “immigration of Tyler,” and “constant value theorem” as “Constantine of doom.” Amusing mondegreens like those make a somewhat tedious task a bit less of a slog for me.

On most episodes of My Favorite Theorem, we talk with one mathematician about their favorite theorem, but our most recent episode is a little different. You can listen to it at kpknudson.com, where the transcript also lives.

Kevin and I went to the Joint Mathematics Meetings in January and interviewed a small subset of the 6,000 or so attendees and asked for their “flash favorite theorems,” super short descriptions of their favorite theorems. We put together an episode with 16 of these “flash favorite” theorems, and it’s a lot of fun, if I do say so myself.

Most My Favorite Theorem episodes don’t have too many distinct mondegreens. The same bits of jargon tend to get interpreted in only one or two different incorrect ways. But a collection of 16 different voices (18 if you count Kevin and me), along with the background noise of a huge meeting, really cranked up the misheard math.

For your amusement, some mathematical mondegreens, courtesy of Otter and our friends at the Joint Math Meetings.

- Cauchy-Riemann equations: cushy remote equations (Eric Sullivan)
- Nullstellensatz: “small scale insights” and “nutshell and thoughts” (Courtney Gibbons)
- Banach-Tarski paradox: “throw the ball not a paradox” and “monitors to paradox” (Shelley Kandola)
- Euler’s identity on the Riemann zeta function: loyalist identity on the remasters function

Riemann zeta function: remain seated function (Terence Tsui) - Math biologist: massive biologist (Courtney Davis)
- Isomorphism: is Marxism (Jenny Kenkel)
- linear algebra: when you’re out there (Charlie Cunningham)
- Voronoi: very annoying (Ellie Dannenberg)
- Sharkovskii: circus be (Michelle Manes)
- Gödel’s incompleteness: Good Al’s inconvenience (John Cobb)

Poor Al! If only someone would make it so the axioms of mathematics had no limitations at all, he’d be so much happier! Have you ever heard a mathematical mondegreen? Let me know on Twitter.

]]>“Your Bowl of Rice Is Hurting the Climate Too” reads a Bloomberg headline from June. “Rice cultivation could be as bad for global warming as 1,200 coal plants, so why aren’t consumers more bothered? Eco-conscious consumers are giving up meat and driving electric cars to do their part for the environment, but what about that bowl of rice?” I was irritated as soon as I read it. It was probably a combination of the whataboutism and the focus on a food that is eaten much more in Asia and Africa than the U.S. and Europe when overall Americans and Europeans have caused a lot more greenhouse gas emissions per capita than Africans and Asians. To top it off, what should I make of the 1200 coal plants number? How much of a climate impact “should” the staple food of billions of humans have?

The article is full of figures. They all sound impressive, but I didn’t really understand how to interpret them. Rice is “just as damaging over the long term as annual carbon dioxide emissions from fossil fuels in Germany, Italy, Spain and the U.K. combined.” (Do Germany, Italy, Spain, and the U.K. all rely on fossil fuels for most of their energy? How do the populations of those four countries compare to the population that relies on rice for a significant proportion of their calories? How should I compare climate impact of the farming of one crop for the entire world to the climate impact from all causes in a few countries?) “Global production of milled rice has increased 230% since 1960.” (How much has the population increased since then?) Rice production emits “twice as much of the harmful gases as wheat.” (Is more rice or wheat consumed?) “Growing rice in flooded conditions causes up to 12% of global emissions of methane, a gas blamed for about one quarter of global warming caused by humans.” (What are the major sources of anthropogenic methane emissions? Methane from rice farming causes 3% of anthropogenic global warming. Is that a lot? Food is one of the least optional sources of greenhouse gas emissions, after all. Plenty of people live without cars, flights, or electricity, but calories are a must.)

I’m not writing this post solely because I wanted to complain about one article that bugged me, as fun as that is. It’s important to think about how we interpret headlines like this one. Many people have had traumatizing experiences with mathematics and don’t feel comfortable reasoning about numbers or statistics, but as a society we are also on the whole deferential to numbers. An article can get away with throwing statistics around without properly contextualizing them because people won’t question them, or don’t know the right questions to ask, or think an argument that refers to a lot of numbers must be a sound one.

Furthermore, humans’ perceptions about what to focus on when it comes to pollution and climate change can be skewed. Recently, some environmental activists have zeroed in on plastic straw waste, causing a reaction from disability activists, who say plastic straws are important accessibility items for some people. The fracas concerns less than a tenth of a percent of the plastic pollution in the ocean. A plastic straw ban is basically symbolic. (A 2018 study estimated that about half of the plastic pollution in the famous Great Pacific Garbage Patch consists of lost or discarded fishing nets.) With a limited mental bandwidth for caring about and taking action on various environmental issues, where should rice fall on that list?

But most of all, I finished reading the article honestly unsure how to understand the impact of rice farming on the environment. I wanted to find the numbers that would help me put the situation in context.

Back to rice.

Reading the article, my first question was what proportion of the world’s calories come from rice. That seems like an important basic fact that would help me understand the other numbers. Ricepedia, a rice information site run by CGIAR, an agriculture research organization, says 19% of “global human per capita energy” comes from rice. About 3.5 billion people get at least 20% of their calories from rice, and about half a billion get most of their calories from rice. Other sources I found had similar numbers, reporting 16–20% for the proportion of the world’s calories that come from rice.

A fifth of the total calories humanity consumes is a lot. Corn (maize) and wheat have similar numbers. Together, the three plants provide more than half of our calories. Of course growing something that sustains so many people will have an impact on the environment. I was somewhat surprised that rice, corn, and wheat were so similar in the proportion of calories they provide. It helped put the fact that rice farming causes twice as much greenhouse gas emission as wheat in perspective.

My next question was how rice’s impact stacks up against impacts from other foods and how that compares to its importance as a source of nutrition. The statistic that rice produces 12% of anthropogenic methane and that the methane produced by rice farming makes put about half of crop-related greenhouse gas emissions come from a white paper prepared by the Environmental Defense Fund (EDF). (The white paper isn’t actually about the methane emissions; it is about a study that shows that attempts to mitigate methane emissions may be increasing the emissions of nitrous oxide, another potent greenhouse gas.)

The EDF bases their estimates on the 5th Assessment Report of the United Nations Intergovernmental Panel on Climate Change (relevant chapter here). According to those numbers (specifically figure 6.8, if you’re following along at home), the main food-related contributors to anthropogenic methane emissions are rice paddies and cow farts. (They don’t quite use that terminology.) Together, those sources account for about 40% of anthropogenic methane emissions, with rice producing about 30% of that amount. If all foods emitted the same amount of methane, rice would only produce 20%, so it produces about 1.5 times as much as it “should” proportionally. But the real story is that the methane emissions of food are very disproportionate, with rice and ruminants almost completely responsible! When all greenhouse gas emissions from food are taken into account, rice emits more greenhouse gases per calorie than wheat or corn but less than fruits, vegetables, legumes, or any animal sources. See this working paper from the World Resources Institute for more granular data. If the EDF is correct that rice emits more nitrous oxide than previously understood, those numbers may underestimate rice’s impact, but it is still dwarfed by the impact of animal-based foods.

I haven’t answered all the questions the article left me with, but I actually feel a lot more equipped to understand the impact of rice on the environment. Some of the numbers from the article still perplex me. I don’t know what to make of the comparison to 1200 coal plants or the assertion about Germany, Italy, Spain, and the U.K. Comparing the climate impact of fossil fuel use in all sectors in four countries to the impact of a food that is eaten all over the world just doesn’t make sense to me. The statistic about rice production increasing since 1960 is a little more meaningful. The world population today is about 2.5 times as much as it was in 1960 (so it has increased 150%). Rice production, though, is about 3.3 times as much, so rice production has grown more than the population by a moderate amount.

Personally, these statistics will probably not change my rice consumption. I don’t eat a lot of rice anyway. It’s a part of my diet, but I get a lot more of my calories from wheat, and I think decreasing my consumption of dairy products would probably be more effective in reducing the greenhouse gas emissions of my diet than reducing the amount of rice I eat would be. More broadly, rice is an important source of nutrition and part of the cultural heritage for billions of people and can be grown in places other crops can’t, so I bristle at any implication that people who rely on rice as a staple should cut down on it or are making irresponsible choices by surviving, and throwing shade at consumers who reduce their meat consumption but not rice seems particularly unhelpful.

That said, the statistics I found about the environmental impact of rice farming did surprise me. I didn’t realize it was such an outlier from other grains in terms of its climate impact, and I am glad that research continues into how to grow rice in less damaging ways. My dive down the rice rabbit hole also highlighted to me how difficult it can be to obtain and interpret information about how our choices affect the environment. Figuring out the full context for the numbers is difficult, and I hope more climate change research organizations will continue to make it easier for everyone to get the information they need to make informed choices.

]]>The Mathematics Genealogy Project (MGP) is a website that collects information about the advisors and students of PhD mathematicians. It can be fun to waste a little time clicking around the site, finding famous ancestors (I’m one of Leonhard Euler’s 113,581 known mathematical descendants) or figuring out who else is in your or your favorite mathematician’s lineage.

There are quite a few women in my mathematical family tree, but none of them are my direct ancestors. That got me wondering about the longest chain of women advisors and advisees. The MGP does not include gender data, but I know the names of some prominent women in math from the 19th and early 20th centuries, so I started exploring.

Sofya Kovalevskaya is considered by most people to be the first woman to earn a doctorate in mathematics (she did so in 1874 from the University of Göttingen), but her only student, Emil Stenberg, was a man. Charlotte Angas Scott, who earned a doctorate in 1885 from Cambridge University, was as far as I can tell the first woman to graduate a woman student. (She was the first woman to have a doctorate in mathematics in America, although she was not the first American woman to have a doctorate in mathematics. She achieved the former by crossing the Atlantic.) Between 1896 and 1925 she graduated eight women math PhD students from Bryn Mawr, but none of them advised doctoral students.

Winifred Merrill was the first American woman to receive a PhD in math, which she did from Columbia in 1886, but again she did not have students. Anna Pell Wheeler, who earned a PhD in 1910, advised six women PhD students at Bryn Mawr. One of them, Dorothy Maharam, had four students of her own, all of whom I believe to be men. Emmy Noether had two women PhD students, Margarethe Hermann and Ruth Stauffer, but neither advised doctoral students.

After striking out on these early mathematical foremothers, I decided to check the Olgas. (In my experience, Olga seems to be an especially mathematical name.) Olga Taussky-Todd’s student Hanna Neumann advised ten students, as far as I can tell all men. There were several women among Olga Oleĭnik’s 48 students listed in the MGP (the MGP is more accurate for American than Russian mathematicians, so 48 may be an undercount), but none of them advised students.

With Olga Ladyzhenskaya, I may have hit the jackpot. Her student Nina Uraltseva has advised 13 students. Two of Uraltseva’s women students, Arina Arkhipova and Nina Ivochkina, have advised women students of their own. In total, there are five advisor-advisee chains of length four containing only women.* Several of these women could still be advising students, so these chains may continue to grow.

I don’t know a good way to search MGP or any other resources for long chains of women mathematicians, so I may have missed some. (Know of a longer chain of women with women advisors? Please tell me about it!) In the end, it’s more of a curiosity than anything else. Focusing only on direct doctoral advisors is a coarse measure; students learn from and are mentored by many others aside from their official advisors on their way to their degrees. And gender is not binary, though our histories usually assume people are either men or women, based on assignments made at birth. I hope that if anyone dusts off this blog post in a few hundred years they will find my fixation on gender confusing, but today I like knowing that among the thousands of chains of men advised by men, there are a few women who form chains of women advisors and advisees.

*After I published the original version of this post, Steve Kass found a chain of length 4 starting with Stella Dafermos, and I believe there are probably several other chains of length 4. Please keep updating me if you find more, and especially if you find a longer chain. The MGP is an imperfect resource, and if you find a chain of people that do not have entries there, you can submit an update.

]]>My favorite headline for this one: “Viral math problem baffles mathematicians, physicists.” I am greatly amused at the implication that large numbers of mathematicians and physicists have abandoned their analytic number theory or quantum gravity to get to the bottom of an ambiguous arithmetic problem.

Questions like 8÷2(2+2) boil down to whether you compute the answer left to right or do the multiplication on the right side of the obelus (division symbol) first. Part of the reason this expression is ambiguous is that some people the acronym American students learn, PEMDAS (parentheses exponents multiplication division addition subtraction), as meaning multiplication comes before division when they should have equal priority. (Acronyms used in some English-speaking countries put the D before the M.) One method yields 1 as the answer and the other 16.

I used to explain that when people sent me messages about some viral order of operations questions, but for the past few cycles (they seem to bubble up every several months like an algal bloom in a lake) I have adopted a strict policy of refusing to answer. I do believe one of the answers is more correct than the other, but I will not tell you which one.

The real answer, the one I believe any mathematician, physicist, engineer, other number-cruncher would tell you is to make sure your expressions aren’t ambiguous. There’s no extra charge for another set of parentheses. Just toss them in. If you want the answer to be 16, write (8÷2)(2+2). If you want it to be 1, write 8÷(2(2+2)). Problem solved. Some people leave school math classes believing math is a minefield studded with gotchas. It’s not supposed to be like that, and it’s a real shame so many people end up with that impression. Part of the job of anyone who writes an expression like that is to make sure it can be understood. If it’s ambiguous and someone gets the “wrong” answer, the blame belongs to the person who wrote the ambiguous question.

The bar is low, but I think the question of why these viral questions keep popping up is much more interesting than what the answers are. Is it cathartic for people who were traumatized by what they viewed as arbitrary rules in math? “See? Math is confusing and ambiguous!” Is it wanting to be right and shame people who are wrong? Maybe it’s the arithmetic equivalent of the dress or Yanny/Laurel: We can all understand the question, think we know the answer, and can’t imagine how anyone could see anything different. Is it just a relief to have passionate internet arguments about something much less important than healthcare or immigration? If you like these questions, what is it you like about them? (You can tell me on Twitter.)

Steven Strogatz has already written about this topic more elegantly for the New York Times. I’ll close by echoing his appeal to those of us who talk about math in public to help people see beyond mathematics as a subject with arbitrary rules and black and white answers to the more interesting questions that motivate mathematicians.

]]>My advisor used to use the phrase “preservation of difficulty.” If I gave up on one approach for solving a problem because I ran into a seemingly impassable difficulty, a new approach was likely to throw an equal but different difficulty my way. Mathematicians do sometimes find real shortcuts or have real breakthroughs that make everything easier, but more often they wear away at difficulties from one side or another until they get somewhere.

Kevin Knudson and I talked with Jim Propp for the most recent episode of our podcast My Favorite Theorem (audio and transcript here), and our conversation reminded me of the idea of preservation of difficulty, though *difficulty* is not quite the right word in this situation.

Dr. Propp talked about a theorem that seems so obvious as to barely be a theorem: If a function does not change, it is a constant. Good job, Sherlock! But then he peeled back some of the layers of this statement to show that an axiom called the completeness of the real numbers lies at the base of the proof of this theorem, which we’ll call the constant value theorem.

The completeness axiom states that there are no gaps in the number line. One way of formalizing the idea is the following statement: Every nonempty subset of the real numbers that has an upper bound has a least upper bound. For example, the set of rational numbers less than the square root of 2 has a lot of upper bounds: 17, 500, π, the list is literally uncountable. But √2 is the smallest one.

As an axiom, the completeness of the real numbers is assumed to be true without proof, at least when you’re doing math on the standard real number line. (Mathematicians love to figure out what happens when axioms are removed or replaced, so not every piece of mathematics takes the completeness axiom for granted.) But it doesn’t have to be. We could assume the constant value theorem instead and derive the completeness axiom as a theorem. We could assume some other properties altogether and derive both the constant value theorem and completeness axioms.

We can do a similar thing with other axioms: make the axiom a theorem by assuming a different theorem as an axiom. When I wrote about this idea with the parallel postulate, I imagined it as a stubborn wrinkle in a sheet. If you smooth it down in one place, it pops up somewhere else. There’s a preservation of, not exactly difficulty, but some kernel of the axiom that must be assumed rather than proved.

Dr. Propp wrote an article called “Real Analysis in Reverse” delving more deeply into this idea, working through some theorems that are equivalent to the completeness axiom and some others that seem similar but are not equivalent. It’s interesting to imagine the advantages and disadvantages of using any of them as an axiom, but the article also gets into what these equivalences and non-equivalences tell us about the real numbers themselves. He writes, “the main theme of this essay is that anything that isn’t the real number system must be different from the real number system in many ways.” Probing exactly what these differences are can help us understand the special domain in which we have chosen to do a great deal of mathematics. For more details, check out our podcast episode, Dr. Propp’s accompanying essay on his lovely blog Mathematical Enchantments, or his article.

]]>As I was drifting off to sleep one night, I had one of those brilliant ideas that only comes along when you’re drifting off to sleep: diagonalizing the psalms. Earlier that day I had noticed that Psalm 119 was very long—longer than 119 verses, in fact—and wondered how many psalms from the Book of Psalms have at least as many verses as their number in the psalter. One thing led to another, and my subconscious decided the idea of applying Cantor’s diagonal argument to create a new psalm seemed incredibly compelling.

Cantor’s diagonal argument is one of my very favorite proofs. I still get goosebumps sometimes thinking about it. It is a proof that the infinity of the counting numbers 1, 2, 3, 4,… is smaller than the infinity of the real numbers, or even the real numbers between 0 and 1. The real numbers are unlistable, or uncountable. The argument goes like this: Suppose you have a way to list the real numbers. That is, you have some infinite string of decimal digits for the first number on your list, a different infinite string of decimal digits for the second number on your list, and so on. You claim that every real number appears somewhere on this list, but Cantor says he will create a number that is not on the list. He builds this new number this way: The first digit of the new number is anything other than the first digit of the first number. The second digit of the new number is anything other than the second digit of the second number, and so on. (It really doesn’t matter what other digit you choose in each place, as long as it doesn’t match.) The Nth digit of the new number doesn’t match the Nth digit of the Nth number on your list.

How do we know the number doesn’t appear on the list? Well, it can’t be the first number on the list because it doesn’t have the same first digit. It can’t be the second number on the list because it doesn’t have the same second digit. You get the idea. You could add the new number to the list somewhere, but Cantor will just produce another number not on your list, and another one after that, as if pulling rabbits out of a hat. (The argument doesn’t work for the list of whole numbers because they have a finite number of digits and doesn’t work for rational numbers because it doesn’t produce a rational number.)

My first idea for diagonalizing the psalms was just to make a diagonal psalm: the first verse of Psalm 1, the second verse of Psalm 2, and the Nth verse of any Psalm N that has at least N verses. With the help of the Book of Common Prayer and this list of the psalms ordered by length, I came up with my first diagonal psalm:

**A First Diagonal Psalm**

Happy are they who have not walked in the counsel of the wicked, *

nor lingered in the way of sinners,

nor sat in the seats of the scornful!

Why do the kings of the earth rise up in revolt,

and the princes plot together, *

against the LORD and against his Anointed?

But you, O LORD, are a shield about me; *

you are my glory, the one who lifts up my head.

Tremble, then, and do not sin; *

speak to your heart in silence upon your bed.

Braggarts cannot stand in your sight; *

you hate all those who work wickedness.

I grow weary because of my groaning; *

every night I drench my bed

and flood my couch with tears.

Awake, O my God, decree justice; *

let the assembly of the peoples gather round you.

All sheep and oxen, *

even the wild beasts of the field,

The LORD will be a refuge for the oppressed, *

a refuge in time of trouble.

The innocent are broken and humbled before them; *

the helpless fall before their power.

He delivered me from my strong enemies

and from those who hated me; *

for they were too mighty for me.

Praise the LORD, you that fear him; *

stand in awe of him, O offspring of Israel;

all you of Jacob’s line, give glory.

I have seen the wicked in their arrogance, *

flourishing like a tree in full leaf.

In your sight all the wicked of the earth are but dross; *

therefore I love your decrees.

It definitely doesn’t have the coherence of the actual psalms, but it hits some of the highlights of the book as a whole: wickedness is bad, animals exist, God protects the faithful. I was a little surprised it only has 14 verses. Only Psalms 1-10, 18, 22, 37, and 119 have enough verses to appear in my diagonalized psalms. There are a lot of psalms that are pretty long, but few of them line up well with their numbers. I was most disappointed with Psalm 78, which nearly makes the list but has only 72 verses. (Would anyone mind terribly if we swapped psalms 72 and 78?)

My first diagonal psalm might be a Cliff’s Notes version of the Psalter, but it’s kind of a shame that a poem born of the observation that Psalm 119 is long is so short. It would be nice if a newly created psalm felt longer. One option would be to insert a pause for every psalm that does not have enough verses. It starts out strong and then peters out quickly.

**A Dramatic Diagonal Psalm**

Happy are they who have not walked in the counsel of the wicked, *

nor lingered in the way of sinners,

nor sat in the seats of the scornful!

Why do the kings of the earth rise up in revolt,

and the princes plot together, *

against the LORD and against his Anointed?

But you, O LORD, are a shield about me; *

you are my glory, the one who lifts up my head.

Tremble, then, and do not sin; *

speak to your heart in silence upon your bed.

Braggarts cannot stand in your sight; *

you hate all those who work wickedness.

I grow weary because of my groaning; *

every night I drench my bed

and flood my couch with tears.

Awake, O my God, decree justice; *

let the assembly of the peoples gather round you.

All sheep and oxen, *

even the wild beasts of the field,

The LORD will be a refuge for the oppressed, *

a refuge in time of trouble.

The innocent are broken and humbled before them; *

the helpless fall before their power.

*(7 pauses)*

He delivered me from my strong enemies

and from those who hated me; *

for they were too mighty for me.

*(3 pauses)*

Praise the LORD, you that fear him; *

stand in awe of him, O offspring of Israel;

all you of Jacob’s line, give glory.

*(14 pauses)*

I have seen the wicked in their arrogance, *

flourishing like a tree in full leaf.

*(81 pauses)*

In your sight all the wicked of the earth are but dross; *

therefore I love your decrees.

*(31 pauses)*

If you want to choose a set duration for the pauses, this would be a perfect way to emphasize how many psalms are being omitted in a dramatic reading of the psalm. If using this psalm as the text for a choral composition, the voices could rest during the pauses while any instruments played.

An option that would allow all psalms to contribute would be to throw in some modular arithmetic. Modular arithmetic is also called clock arithmetic because it treats numbers as if they wrapped around in a cycle. Five hours after 10:00 is usually called 3:00, not 15:00, because we reset from 12 to 1. To make a diagonal psalm this way, suppose Psalm N is M verses long. The Nth verse of this diagonal psalm is the N (mod M) verse of Psalm N. The first 10 verses of this psalm are the same as my first diagonal psalm. Then we get to Psalm 11, which has only 8 verses. Eleven is 3 mod 8, so we add the third verse of Psalm 11. Psalm 12 also has 8 verses, so we’d use the fourth verse in our psalm because 12 is 4 mod 8. At 150 verses, this diagonal psalm is still shorter than Psalm 119! (But long enough that you’ll need to click “Show/hide” below to expand.)

**A Modular Diagonal Psalm**

Although they were inspired by late-night ruminations on Cantor’s diagonal argument, none of these psalms actually employs a key element of Cantor’s argument, changing some aspect of the nth entry. Even if the purpose of the diagonal psalm were to create a psalm not in the Psalter, we would not require this technique. It’s pretty easy to see by inspection that none of the versions we’ve made match any already-existing psalm. But as this exercise was never one of practicality, I decided to write one last psalm based on this part of Cantor’s argument. There are a lot of ways to alter the nth verse of the nth psalm. I decided to incorporate the N+7 idea from French experimental literary group Oulipo. This psalm is created by replacing the first noun in the nth verse of the nth psalm by the noun 7 after it in a dictionary. Because I do not actually own a physical dictionary and online dictionaries are not easy to browse, I used this online N+7 generator. (Its dictionary doesn’t have “arrogance,” so I browsed an online dictionary to replace that one.)

**An Oulipian Cantorian Diagonal Psalm**

Happy are they who have not walked in the counterbalance of the wicked, *

nor lingered in the way of sinners,

nor sat in the seats of the scornful!

Why do the kips of the earth rise up in revolt,

and the princes plot together, *

against the LORD and against his Anointed?

But you, O LORD, are a shingle about me; *

you are my glory, the one who lifts up my head.

Tremble, then, and do not sin; *

speak to your heartthrob in silence upon your bed.

Branches cannot stand in your sight; *

you hate all those who work wickedness.

I grow weary because of my groaning; *

every nightlight I drench my bed

and flood my couch with tears.

Awake, O my God, decree kayak; *

let the assembly of the peoples gather round you.

All shelves and oxen, *

even the wild beasts of the field,

The LORD will be a regatta for the oppressed, *

a refuge in time of trouble.

The innocent are broken and humbled before them; *

the helpless fall before their praise.

He delivered me from my strong Englishmen

and from those who hated me; *

for they were too mighty for me.

Praise the LORD, you that fear him; *

stand in babble of him, O offspring of Israel;

all you of Jacob’s line, give glory.

I have seen the wicked in their arrowsmith, *

flourishing like a tree in full leaf.

In your signature all the wicked of the earth are but dross; *

therefore I love your decrees.

Because the first noun of the first verse is different from the first noun of Psalm 1, we know this is not Psalm 1. Because the first noun of the second verse is different from the first noun of Psalm 2, we know it is not Psalm 2. And so we have triumphed. My dream of a Cantor-style diagonalized psalm has been made reality!

For another exploration of algorithmic religion, check out the Flash Forward podcast episode “Our Father, who art in algorithm.“

]]>A friend who recently defended his dissertation in comparative literature mentioned Simone Weil’s writing on the *Iliad* in his defense. Afterwards, I told him her brother André was a famous mathematician. (In my former field of research, his name adorns the frequently-referred-to Weil-Petersson metric, but he is also a household name in other fields of math for other reasons.) My friend was unfamiliar with him. I myself had only learned of Simone a few years ago in Francis Su’s poignant lecture “Mathematics for Human Flourishing,” which repeatedly circles back to her words: “Every being cries out silently to be read differently.”

Both Weil siblings were intense and devoted to their work. They were close and loved each other, but their ways of moving through the world were starkly different. Simone was a philosopher and political activist, dedicated to the struggle of the common person. Alongside her dense philosophical writing, she was drawn to manual labor and wanted to suffer with those who were suffering. She died at age 34 in the middle of World War II, possibly as a result of her refusal to eat more than she believed children in German-occupied France had for rations after she contracted tuberculosis. André was a bold, often abrasive mathematician who traveled extensively and eventually married a woman who was the wife of a colleague when they met. He was arrested in Finland on suspicion of being a Soviet spy near the beginning of the war and held in prison before leaving to join the French army briefly and then emigrating to the U.S. after Germany occupied France. He died in 1996 at age 92.

Karen Olsson paints vivid portraits of both siblings in her forthcoming book The Weil Conjectures*. *With it, she invites the reader to sit with the Weils, to appreciate their relationship and ponder what their lives and work say to contemporary writers and mathematicians.

The book is not a biography of either Weil or a detailed look at any of André’s math, which was what I was expecting to some degree based on the title. It is more impressionistic than that, with Olsson weaving other historical vignettes and her own relationship to math and writing in with the story of the Weils. Olsson was enamored of mathematics for a few years in college and flirted with the idea of going to graduate school for math before deciding she wanted to be a writer instead. (In an amusing passage, she writes about pestering a friend “to admit that he hoped to write a novel eventually, which I believed everyone secretly did.” Eventually his insistence that he really wanted to be a mathematician and not a novelist helped her understand that her desire to write was not universal and that perhaps she should pursue writing seriously.)

But throughout the book she writes about revisiting the subject decades after her last college math course, watching abstract algebra lectures from a Harvard class as a refresher. She is not sure why she is pulled to the subject again so strongly but observes similarities between writing and mathematics. “How I would like to write something as clean and powerful as the best kind of mathematical proof,” she writes. Later: “A quality of both good literature and good mathematics is that they may lead you to a result that is wholly surprising yet seems inevitable once you’ve been shown the way, so that—aha!—you become newly aware of connections you didn’t see before.” As a mathematician-turned-writer who took a different trajectory than Olsson did, I was interested in our similarities and differences in our stories and feelings about math and writing. (I was amused by her recollection of something she felt in college: “I want to be a *real* writer. I’m not going to write about *math*.” Sick burn!)

Simone and André butted heads about mathematics, though both were interested in it. Simone thought mathematics was too abstract and irrelevant to the life of the common person, André was dismissive about explaining mathematics to non-mathematicians, describing it as “explaining a symphony to a deaf person.” But he did try to explain his work to his sister

The book feels deliberately fragmentary. Olsson will spend a few paragraphs with Simone or André, move to the other abruptly, and then switch to another piece of mathematics or history, or her own story. The quick changes can cause a little bit of whiplash and could probably have been deployed a bit more sparingly. On the other hand, some of her insights about the process of writing or mathematics are sharper for having been juxtaposed so closely with relevant parts of the Weils’ story.

After reading the book, my overwhelming thoughts (though I don’t know if it was Olsson’s intent) are of the form “what if?” What if Simone had survived the war? What if the Weils had been born at a different time, when the war would not have interrupted (or ended) their lives? What if André had been a more compassionate person? What if Yutaka Taniyama, a younger Japanese mathematician who helped formulate an important conjecture in number theory known as the Taniyama-Shimura-Weil conjecture or the modularity theorem now that it has been proved, had not died by suicide so young? (Olsson writes about him and his colleague Goro Shimura, who died earlier this year.) I think of all the paths not taken. Olsson and mathematics. Me and music. You probably have your own. We all make the choices we think are right given our circumstances, but what if things were different?

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