Of birds and wires

Leonard Cohen died on November 7, 2016. He was very popular in Poland in the 1970s and 80s, long before Hallelujah, before the world tours and the late commercial success. We loved our obscure-not-obscure artists, even as we misunderstood or misinterpreted them. We mispronounced his name (“Lee-oh-nard”). We didn’t understand English well enough to get the wry sense of humour or the sexual innuendos. And still.

We had no commercial radio at the time, and no record industry to speak of. Western music was brought to us by enthusiasts who travelled abroad – not many of us could – and spent their own money to buy records, then played them in clubs or on the radio. The rest of us made mix tapes off radio broadcasts, borrowed records and tapes from those who had access to them, stayed up late or rearranged our schedules to listen to music we cared about. There was no Western style commercial promotion through exposure. There was institutionalized political pressure to play Soviet bloc artists, but few, if any, commercial incentives to promote Western rock music. The DJs and broadcasters played it because they loved it, and the audience listened because we loved it back.

Cohen’s fandom first percolated to Poland through word of mouth: a borrowed record here, a tape there. Then a dude, Maciej Zembaty, translated some of Cohen’s songs into Polish and started singing and recording them. It took off like wildfire.

It was not all Cohen all the time, of course. We listened to Led Zeppelin and Pink Floyd and the Beatles, and Hendrix and Tangerine Dream and Dead Can Dance. They were beloved, but also intimidating. You could blast Zep II or Tubular Bells or The Dark Side Of The Moon on your home stereo equipment and get blown away by the sound effects. You could delve into the complexities of The Wall. But when we needed something to sing around the campfire, or on a train, or in a dorm room when a conversation was too much and silence was not enough, few of us would attempt Floyd or Zep. Maybe some of the ballads, and even that was hard.

Cohen was more forgiving. It was OK if you only had a cheap guitar. It was OK to sing Cohen badly; after all, he was doing that himself. Your back could be bent into a permanent question mark, your lungs shrivelled and throat inflamed from the coal dust or chemical pollution or cigarette smoke. You could be missing a few teeth, as people often do when the food does not nourish, hygiene is impossible to maintain, and dentistry is the stuff of nightmares. You could still sing Cohen. And that might have been because he, as the songwriter, had done most of the heavy lifting for you in advance. Bob Dylan, interviewed for a New Yorker article, praises Cohen’s musical gift:

When people talk about Leonard, they fail to mention his melodies, which to me, along with his lyrics, are his greatest genius… Even the counterpoint lines—they give a celestial character and melodic lift to every one of his songs. As far as I know, no one else comes close to this in modern music. … [Cohen’s] gift or genius is in his connection to the music of the spheres.

For all of Cohen’s self-deprecating comments about his “golden voice,” he wrote melodies that were eternal and indestructible. They could withstand all the abuse that we inflicted on them, the drunken performances, the missing chords and forgotten lyrics. It would still be alright.

He was forgiving in other ways as well. I learned later that, in the land of the constitutionally guaranteed pursuit of happiness, Cohen was considered dark and depressing. That was not how we saw it. Sure, he sang of broken people, failed promises, lost wars. These were statements of facts that were just true, even when we did not have the permission or ability to say so. Having them spoken out loud felt like understanding and forgiveness. It might have even felt uplifting, in telling us that such things mattered, that they were worth a song.

On November 9, 2016, some Americans woke up feeling that they were in a country they did not know. Disoriented, they looked to historians and philosophers of faraway places for advice and consolation. They resolved to remember what normal life looked like and take note of everything that was not normal. They made lists of things they would not do and compromises they would not make.

Oh, you sweet summer children. I do hope that you will act, that your institutions can be mobilized to prevent the worst. I really do, for your sake and my own and that of everyone else on the planet. But since you ask me so often where I’m from, let me tell you what it’s like to live under oppression and see no end of it.

Continue reading “Of birds and wires”

Art in the life of mathematicians


This book has been in the works for some years now, and I’m thrilled to finally have a demo copy to show you. The book will be published by the American Mathematical Society. The demo copy has been produced (impressively quickly!) by the Hungarian publisher Ab Ovo. I’m very grateful to Anna Kepes Szemerédi for envisioning this project in the first place, and for all the hard work she has put into it.

I have contributed an essay on photography. You can download it here, and here is the gallery of the photos I offered to be used in the book. The photo on the cover is also mine. I hope that this will encourage you to purchase the book when it becomes available; I’m only one out of many contributors (see the cover for the list of names), and the book format will add further value through graphic design. If you’re expecting “mathematical art” as exemplified for example by the Bridges conference, I must warn you that this is not what I do. (In the essay, I explain why.) There is some overlap with one of my blog posts from last year: the post was adapted from an earlier version of the essay, and then I used it in writing the final version.

Anna first approached me about this in late 2011. I was much less confident then, both in my photography and in my writing. I have worked on both since then. One thing I wish I’d seen before I submitted my contribution is this classic piece by Linda Nochlin on the absence of great women artists in the history of art. Here’s what she says about “the lady’s accomplishment”:

In contrast to the single-mindedness and commitment demanded of a chef d’ecole, we might set the image of the “lady painter” established by 19th century etiquette books and reinforced in the literature of the times. It is precisely the insistence upon a modest, proficient, self demeaning level of amateurism as a “suitable accomplishment” for the well brought up young woman, who naturally would want to direct her major attention to the welfare of others–family and husband–that militated, and still militates, against any real accomplishment on the part of women. It is this emphasis which transforms serious commitment to frivolous self-indulgence, busy work, or occupational therapy, and today, more than ever, in suburban bastions of the feminine mystique, tends to distort the whole notion of what art is and what kind of social role it plays.

This got me thinking back on what I wrote about photography and wondering for a moment if I might have fallen into the trap of “suitable accomplishment.” In the end, it clarified for me the distinction between the commitment to the process of getting better, and the expectation of achieving a certain level of excellence, and the expectation of gaining public acclaim. I have always been anything but unambitious. Nonetheless, I have never aimed to be a “great artist.” I am not altogether indifferent to success in art, as evidenced by this self-promotional post, but what made me pick up the camera is the pleasure I find in taking photographs. My enjoyment of it is not conditional on finding an audience, receiving public recognition, or on any presumption of greatness. Instead, it comes from trying to get better at it. The pleasure is not in taking the same photographs over and over again, but in expanding my range, improving my technique, seeking out new ideas and solutions. The seriousness of my commitment is in my engagement in the process.

I suppose that this does not make me a lady.

G.H. Hardy and Mrs. Ellis

David, by Michelangelo. Image: Wikimedia
David, by Michelangelo. Image: Wikimedia

If you haven’t yet read this classic essay by Linda Nochlin on the question of why there have been no great women artists, I recommend it very highly. The essay is from 1971, but Nochlin’s points remain very much relevant to today’s arguments about why there have been so few great women philosophers, or mathematicians, or whatever.

Nochlin starts out by questioning the common notion of a “great artist” as a singularity that exists independently of society and history. The truth is, it takes at least a village. Great artists are enabled by the society they live in, draw on its artistic traditions, engage in a dialogue with other practitioners. Indeed, if artistic greatness depended only on innate talent, it would be very difficult to explain what Nochlin calls “conditions generally productive of great art,” such as must have existed, for instance, in the 15th century Florence and Rome, or in France in the second half of the 19th century. (We’ll note here that much of the same can be said of mathematics.)

The society also establishes standards for what qualifies as “great art,” and what does not. In the pre-impressionist Europe, historical painting– understood broadly so as to include biblical scenes, Greek and Roman mythology– was considered the highest and most prestigious form of art. Landscapes, still-lifes, portraits, and other suchlike were deemed less worthy. To wit:

Until the 20th century, Mona Lisa was one among many and not the “most famous painting” in the world as it is termed today. Among works in the Louvre, in 1852 its market value was 90,000 francs compared to works by Raphael valued at up to 600,000 francs.

“Great art,” going back to ancient Greece and Rome and then again starting with Renaissance, more often than not depicted naked and partially naked human bodies. Think Michelangelo, Raphael, Titian, Botticelli, Rubens. Even when the figures are clothed, the paintings still display a thorough knowledge of human anatomy. Such knowledge was usually gained through extensive study of the nude model, a practice that continues to be a mainstay of art programs. And yet, as Nochlin explains in detail, nude models (both male and female) were forbidden to women painters before the end of the 19th century. That right there explains completely why there has been no female Michelangelo or Raphael.

Nochlin cites many other ways in which the society refused to enable women artists: the apprenticeship system, access to academic educational institutions such as the Ecole des Beaux-Arts, opportunities to establish suitable relationships with art patrons, and more.

But the part I want to highlight here is the prevailing attitude to “the lady’s accomplishment”:

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Art like science

At the time I was attracted to pure science — physics — where you could speculate and be creative. It’s equivalent to being an artist. If you get the chance, and the cards fall right, there’s no difference. The intellectual play and spirit are the same.

— David Byrne (interviewed by Timothy Leary), 2000


I’ve commented more than once here on the myth of the Mad Scientist: contrary to popular belief, there are no easy shortcuts to scientific greatness. It’s true that some of our creative processes are subconscious, that we sometimes come up with ideas on vacation or after a good night’s sleep. No one, however, becomes a great scientist by just sleeping a lot. Our subconscious faculties only become engaged after we’ve studied the problem and thought about it extensively, often to the point of exhaustion. They don’t kick in every time, and when they do, their input is not even always useful. (I’ve woken up many times with shiny new ideas that did not hold up on inspection.) Excitement, inspiration and quality vacationing can make it easier to put in the sustained, disciplined work of constructing correct and complete mathematical arguments, but does not replace it. As for the relation to actual mental illness, I’ve linked before to a relevant interview with John Nash.

I didn’t get any disagreement on that from math and science types. We understand well enough how the creative process works. We know that being all fired up to prove the Riemann hypothesis is different from actually doing it. Imagine my surprise, then, when I attended a discussion on art and science in the “Philosopher’s Cafe” series a few weeks ago. Scientists and mathematicians came in good numbers, and many of them professed exactly the same kind of misconceptions about art that they would dismiss outright with regard to science. Art, if you believe them, is all about feelings. When a work of art evokes strong emotions, we assume that the artist was overwhelmed accordingly at the moment of creation, leaving no room for intellectual mediation or for calculated, deliberate activity. In other words, the artist experiences an intense feeling, whips himself into a state of rapture, and bang, a painting or whatever materializes in front of him in a puff of magic dust. The Mad Artist swipes his cape and takes a bow, in all his fictional glory.

I’m a research mathematician of some renown. (The regulars here know that, but I’ll say it explicitly anyway, for those who might find this post via links and google searches.) I’m also an amateur photographer (see my Google+ page for samples), and I’ve been attracted to visual arts all my life in some way or other. I’m finding in my own practice that the creative processes in art and in mathematics are often more similar than it might first appear, and I’ve had plenty of confirmation of that from both sides of the aisle. This post is about that, with emphasis on the mathy and sciencey side of art. (Time permitting, there will also be a follow-up post in the converse direction.)

This is not a post about “mathematical art.” Honestly, I have little interest in most of it. I write research papers about fractals, but I find neither mathematical insight nor artistic value in the rainbow-coloured pictures of fractals usually found at math art exhibitions. Don’t even think about sending me links to math rap songs, either. I don’t need art to talk to me about mathematics. I want it to speak to me as art, on its own merits, with no special bonus points for math themes or content.

I’m interested in the less obvious but more organic similarities on the level of the creative process. I’m hardly the first to observe them. Just last year, I attended an artist talk where a painter spoke of his work in terms of “solving the mathematical equation.” Yet, it was plainly in evidence in that discussion a few weeks ago that too many scientists think of art as a softer, lower grade kind of creative endeavour where the concepts of logical thinking and problem solving are pretty much unknown. In that regard, here are a few points to consider.

I’ll be talking mostly about photography, and to some extent about painting, because that’s what I know best. If you think it’s different in other arts, I’ll refer you to Ursula Le Guin’s excellent description of a physicist’s creative process in The Disposessed; I can’t find a link now, but I recall reading somewhere that it was based on her own experiences with writing. If you think that it’s just me thinking that way, that’s very easy to check. There are many artists out there who have blogs, public Facebook or Google+ pages. They might post pictures of work in progress, talk about their influences, recount how a particular piece came about. They might be using different, less “scientific” language, but you will still find a good deal of premeditation, problem solving and analytic thought in what many of them do. And if you tell me that not all art is that great… well, yeah. Not every math paper is a towering pinnacle of intellectual achievement, either. We all do what we can.

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Small expectations

The Vancouver Art Gallery is currently showing Waste Not, an installation by the Chinese artist Song Dong in collaboration with his mother, Zhao Xiangyuan. The exhibition lays out all the items that Zhao collected and hoarded throughout her life. It takes several large rooms to accommodate her treasures: the old clothes, some folded up in neat bundles and others rolled up and tied with ribbons, the shoes sprawled all over the floor, the toys and handbags, the flower pots, plastic baskets, wicker baskets, pots, pans and kitchen utensils, handyman’s tools, cardboard boxes, plastic food containers, styrofoam containers, paper bags, empty bottles and jars, plastic bags folded in small triangles. (You must see the photos here if you haven’t seen the show.)

You might wonder about the mindset of someone who has made it her life’s work to hoard used plastic bags. According to the program notes, Zhao never threw anything away, even if it was broken or no longer usable. She’d lived through shortages of just about everything, and in any case new articles cost money, and anything could potentially, hypothetically, prove useful again.

As you enter the exhibition, the first thing you see is Song’s and Zhao’s notes posted on the wall side by side. Zhao explains how laundry used to be done in China, how ingenuity and folk wisdom minimized the amount of soap needed to keep the clothes clean, soap being rationed and in extremely short supply. (You can read it here if you click on the “artist book”.) Curiosity takes the better of you, so you look around, and the entrance is indeed flanked by a stockpile of coarse soap bars, laundry accessories, detergent boxes and empty cleaning liquid containers.

You could be tempted to jump to conclusions – “those poor people” as well as “why is this supposed to be art?” – but the show encourages you to take more time to think about it. Some small part of your mind has already started wondering about those mysterious little boxes stacked next to the laundry supplies. Toiletries, Chinese medicines, or something else altogether? And that’s Belgian chocolates over there, so I guess they did have those in China? And what the hell is that weird metal thing?

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Sucking at everything else

The June/July issue of the Notices of the AMS features an interview with Gioia De Cari, a former graduate student in mathematics who quit somewhere along the way, went on to become an actress and a playwright instead, and recently wrote and performed a one-woman show about her mathematics experience.

That could have been me, perhaps, in a parallel universe where my graduate student self wasn’t a recent immigrant and had enough of a safety net to be able to contemplate a change of career. Or in another one where I was stuck in Poland instead of going to graduate school in Canada. Or if I had not been available or willing to make several long-distance moves before settling down, or if the only tenure-track academic jobs I could get had been in places where I did not want to live. Even the timeline is close. De Cari was a graduate student at MIT in the late 1980s. I started graduate school in Toronto in 1989.

There would have been a small issue involving my acting skills, or more accurately a lack thereof. Still, I could imagine having had a career in the arts instead, or humanities, or something else with little connection to mathematics. I certainly have thought about quitting mathematics, often and extensively at times, especially in the early years when I was less invested in it. And it’s not like I’ve never had any other interests. At one point, back when I was an undergraduate, I briefly entertained the idea of getting a second degree in the humanities. It was not practical to go ahead with it.
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La Sagrada Familia and the hyperbolic paraboloid

I’m travelling in Spain this month – mostly for mathematical reasons, but, well, it’s Spain. Last week I was fortunate to see La Sagrada Familia.


La Sagrada Familia is the opus magnum of the great Catalan architect and artist Antoni Gaudí. Gaudí was named to be in charge of the project in 1883, at the age of 31, and continued in that role for the rest of his life. From 1914 until his death in 1926 he worked exclusively on the iconic temple, abandoning all other projects and living in a workshop on site.

The construction is still in progress and expected to continue for at least another 20-30 years. The cranes and scaffolding enveloping the temple have almost become an integral part of it. That’s not exactly surprising, given the scale and complexity of the project together with the level of attention to detail that’s evident at every step. Almost every stone is carved separately according to different specifications. Here, for example, is the gorgeous Nativity portal. (Click on the photos for somewhat larger images.)


To call Gaudí’s work unconventional would be a major understatement. To call it novelty – don’t even think about it. His buildings are organic and coherent. Everything about them is thought out, reinvented and then put back together, from the overall plan to the layout of the interior, the design of each room, the furnishings, down to such details as the shape of the railings or the window shutters with little moving flaps to allow ventilation.

Gaudí’s inspiration came from many sources, including nature, philosophy, art and literature, and mathematics.


Continue reading “La Sagrada Familia and the hyperbolic paraboloid”

Maurice Cullen at the Vancouver Art Gallery

If you have a chance to stop by the Vancouver Art Gallery before April 19, there’s a fantastic collection of paintings by Maurice Cullen and his fellow Canadian impressionists on the first floor. The exhibition is subdivided into three parts: Maurice Cullen and His Circle, Maurice Cullen: Intimist, and Exploring Light: Canadian Landscapes from the Permanent Collection.

Maurice Cullen was born in 1866 in St. John’s, Newfoundland, and grew up in Montreal. In 1889, he moved to Paris, France, where he studied painting with Jean-Léon Gérôme and Élie Delaunay. He was clearly influenced by the impressionist movement, then at its height, and learned the impressionist style and techniques.

Now, that might sound like a straight path to becoming a lesser painter making a living off derivative works. After all, you can’t beat Monet or Renoir at their own game. You’re not even likely to be able to add much to their work, unless you’re, say, a Cézanne. Your Monet imitations may still pay your bills, but, well, artistically it won’t get you that far.

Then, however, Cullen moved back to Canada. And, to an impressionist, Canadian landscapes were an uncharted territory waiting to be discovered.

Winter Evening, Quebec

It doesn’t get cold and wet in France like it does in Quebec. Nor do they have snow in France like they do in Canada, covering everything in sight as far as you can see, several feet deep. There’s the early November snow in Quebec: two feet on the ground already, even though the oaks and maple trees haven’t lost their foliage yet. The late December snow: red and purple on a mountainside, illuminated by the early afternoon sunset. The April snow: the more recent snowfall covering the old hardened snow, the blue and green ice floes coming down a river, and all the while the sun is up high and the signs of spring are everywhere.

The French impressionists explored light and nature like no one before them. Painting Canadian landscapes, majestic, austere and beautiful, takes it in a whole new direction. It’s nowhere as apparent as in the Intimist collection, a set of 12 absolutely gorgeous nature paintings by Cullen. St. Catherine Street in Montreal, on a rainy and gloomy November evening, long before it was the busy commercial street it is today. The Laurentians, several times: after an early snowfall, in the midst of harsh winter, in the April sun. A summer landscape, but you can tell that it’s a cool and guarded summer, nowhere close to, say, the Mediterranean heat of southern France.

It wasn’t exactly obvious, before Cullen and his circle of fellow impressionists, that one should even try to paint Canadian landscapes. The attitude of the day was summarized by a quote posted at the exhibition, along the lines of “it’s bad enough that we have to live here, there’s no need to have pictures in the living room to remind us of it”. (I’m quoting from memory.) Cullen, though, found both incredible beauty and an artistic challenge out there. He met the challenge and made his audiences admire the stark, cold, larger than life Canadian nature.

Go here, here, or here for more images. Better yet, see the exhibition if you can.