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.

What if mathematicians wrote travel articles?

Some time ago I suggested that scientists might not always make the best writers. I guess I wasn’t the only person ever to make this profound observation. Slate has since published this piece on how political scientists would cover the news; see also here. As hilarious as these are, I would say that there’s more to the picture. The story below is inspired by this one (hat tip to Terry Tao). Believe it or not, there are actual reasons why we have to write like this sometimes. I’m as guilty as anyone. In fact, I’m in the middle of revising one of my papers right now…

In this article we describe the plane flight that Roger and I took to San Francisco. The purpose of our trip was to meet Sergey, our collaborator on the paper “The structure of fuzzy foils” (J. Fuzzy Alg. Geom. 2003) who also co-organized with me an MSRI workshop in 2005. Our main result was to arrive at the San Francisco airport at the expected time and meet Sergey there. To accomplish this, we relied on a regularly scheduled flight on a commercial airline. For the history of aviation (including commercial aviation) and the general background, we refer the interested reader to Wikipedia (see also Britannica).

This article is organized as follows. We first explain a few preliminary steps, including the travel to the airport and the check-in procedure. The main part of the trip was the actual flight, which we discuss in a new paragraph. We conclude with a few remarks on arriving at the destination airport.

More on mathematics and madness

In popular movies, a scientist is usually brilliant but troubled. We know that he’s brilliant because we’re told so repeatedly, and we know that he’s troubled because that’s plain to see. He might spend a lot of screen time getting depressed over his lack of creative output and trying to remedy this situation by getting drunk or going out for long walks – anything that will keep him from attempting any actual work. Finally, thanks to divine inspiration, a life-changing event or some other such, he stumbles upon a Great Idea. Now that he’s made his breakthrough, the days and nights go by in a blur as the work flies off his hands, the manuscript pages practically writing themselves. Once it’s all done, the scientist has to snap out of the trance, at which point it’s not uncommon for him to collapse and have a nervous breakdown.

I don’t even want to name specific movies – that’s shooting fish in the barrel. The number increases further if you substitute a writer or artist for a scientist. If you’ve seen too many Hollywood films and don’t know better from your own experience, you could be excused for drawing the conclusion that it’s somehow the mental illness that’s responsible for our creativity. I mean, scientific discovery – not to mention art – boils down to blinding flashes of brilliance, and those come hard and fast when you’re seriously kooky, right?

And now there’s a medical study that I’m sure I’ll see quoted in support of this. According to a recent article in Science Daily, researchers at Karolinska Institutet have shown that highly creative people and people with schizophrenia have similar dopamine systems. That in turn has been linked to the capacity for what the article calls “divergent thought” (a scientist is quoted to refer to it as “thinking outside the box”, one of the most annoying phrases out there), which contributes both to creative problem solving in healthy people and abnormal thought processes in people with schizophrenia. The long suspected connection – make sure to also check the links under “related articles” – may thus have a basis in brain chemistry. Yay for the Mad Scientist!

When the truth is gone…

(Updated below: a link added.)

The main ingredients are simple: a house cat, a large box with an airtight lid, a radiation detector, a radioactive sample, and a container of poison gas such as cyanide. Start by placing the poison gas container inside the box and hooking it up to the radiation detector so that if radiation is detected, the gas container is opened and the gas is released into the box. Place the radioactive sample somewhere near the detector; the sample should be chosen so that there is, say, 50% probability that radioactive decay will be observed within an hour. Now put the cat in the box, close the box, leave the room, and shut the door behind you.

According to the superposition principle in quantum mechanics, a radioactive particle does not simply wait a while and then decay at a time chosen randomly according to a given probability distribution. Instead, it evolves into a superposition of a decayed and non-decayed state, and remains so until we check on it by performing an observation. We know that such superposed states must exist, but we never get to see them. The act of taking a measurement causes the particle to actually assume one of the two definite states, either decayed or not, with certainty. A physicist would say that the wave function of the particle collapses upon observation.

But what about the cat? If at least one of the radioactive particles decays, the poison gas is released and the cat dies. Otherwise, the cat survives. You will find out what happened once you open the box. Until then, you’re the proud owner of Schrödinger’s cat: alive and dead simultaneously, a quantum superposition of a live cat and a dead cat as dictated by the wave function of the radioactive sample.

On the other hand, if you would rather keep the kitty away from dangerous contraptions and settle for the philosophical exercise instead, you could do worse than renting A Serious Man, the latest Coen brothers movie.

Putnam

This has been my first year on the Putnam committee: the committee that selects the problems for the William Lowell Putnam undergraduate competition. The committee consists of 3 members appointed for a 3-year term each (each year, one person’s term ends and another one is appointed in his place) and a fourth person, Loren Larson, who is a “permanent” secretary of the committee. To start with, each committee member proposes some number of problems (normally, at least 10). The problem sets and solutions are then circulated and discussed, and eventually the committee meets in person to decide on the final selection. This is all done in strict confidence and well in advance of the actual competition.

I have never written the Putnam. I wrote the Math Olympiad back in the days and qualified for the International Math Olympiad in my last year of high school, but Putnam is not available in Europe. I’m not sure that I would have been interested anyway. I wanted to study the “serious” mathematics: the big theories, the heady generalizations, the grand visions. Olympiads and competitions faded into the distant background and pretty much stayed there until last year.

I did point out my Putnam virginity when I was approached about joining the committee, and was told that Putnam does try to engage from time to time people who are not normally on the circuit, if only to have a larger pool of potential ideas. Of course, the advantage of having people on the committee who are on the Putnam circuit is that they know what’s expected, what works and what doesn’t, what has already been used and shouldn’t be recycled, and so on. Last year’s other two committee members – Mark Krusemeyer and Bjorn Poonen – are Putnam veterans, and of course Bjorn is a four-time Putnam fellow. Mark’s term ends this year; I don’t know yet who will be joining us this January.

Well, you could call it a steep learning curve. Putnam problems are expected to be hard in a particular way: they should require ingenuity and insight, but not the knowledge of any advanced material beyond the first or occasionally second year of undergraduate studies, and there should be a short solution so that, in principle, an infinitely clever person could solve all 12 problems in the allotted 6 hours. (In reality, that doesn’t happen very often, and I’ve heard that it generates considerable attention when someone comes too close.) The problems are divided into two groups of six – A1-A6 for the morning session and B1-B6 for the afternoon session – and there is a gradation of the level of difficulty within each group. A1 is often the hardest to come up with – it should be the easiest of the bunch, but should still require some clever insight and have a certain kind of appeal. The difficulty (for the competitor, not for us) then increases with each group, with A6 and B6 the hardest problems on the exam. There are also various subtle differences between the A-problems and B-problems; this is something that I would not have been aware of if another committee member hadn’t pointed it out to me. For example, a B1 could involve some basic college-level material (e.g. derivatives or matrices), but this would not be acceptable in an A1, which should be completely elementary.

The competition is taking place in two weeks, so you’ll know soon enough what problems we ended up selecting. Meanwhile, it might entertain you to see a few of my duds: problems I proposed that were rejected for various reasons. They will not be appearing on the actual exam and I’m not likely to propose variants of them in the future. The solutions are under the cut, along with an explanation of why each problem is a dud.

1. A ball is shot out of a corner of a square-shaped billiard table at an angle to the edge . The ball travels in a straight line without losing speed; whenever it hits one of the walls of the table, it bounces off it so that the angle of reflection is equal to the angle of incidence. Find all values of such that the ball will hit one of the corners after bouncing off the walls exactly 2009 times.
2. Are there integer numbers such that ?
3. Given , determine the largest integer with the property that any points on a circle must determine at least obtuse angles .

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.

Truth be told:

Yvan Saint-Aubin in the CMS Notes, on behalf of the new bilingualism committee:

Truth be told, writing an elegant, masterful scientific article is possible only when we do so in our mother tongue.

Right. You could tell that to Hoory, Linial and Wigderson, the winners of the 2008 Levi Conant prize for the best expository article in the AMS Bulletin or Notices. Or you could read up on Joseph Conrad, who grew up in Ukraine, Russia and Poland and only became fluent in English in his twenties. Of course, Conrad was only writing fiction, which must be way easier than writing a “masterful” scientific article.

Don’t judge too quickly what others might or might not be capable of.

Update: Emmanuel Kowalski points out in comments that a more accurate translation of the original French would be:

Writing scientific texts elegantly is something that can probably only be done in one’s native language.

Which I still disagree with, but it doesn’t grate like the English version does. I have also removed the sentence that used to be at the top of this post, because I don’t think I would have made good on it.

Best of the best?

The Wall Street Journal is really on a roll, reporting on a Jobs Rated study that names “mathematician” as the best job in the U.S.:

According to the study, mathematicians fared best in part because they typically work in favorable conditions — indoors and in places free of toxic fumes or noise — unlike those toward the bottom of the list like sewage-plant operator, painter and bricklayer. They also aren’t expected to do any heavy lifting, crawling or crouching — attributes associated with occupations such as firefighter, auto mechanic and plumber.

The study also considers pay, which was determined by measuring each job’s median income and growth potential. Mathematicians’ annual income was pegged at $94,160 [...]

The complete ranking of 200 jobs is here, and here is the explanation of the point system on which the ranking was based.

The bottom line is, all point systems should be viewed with a healthy degree of skepticism.

I want to make it very clear that I’m not whining here. I like my job. But it’s not at all for the reasons just mentioned. Not because we spend a lot of time indoors – which, incidentally, can be quite unpleasant if you have a windowless office (I did as a graduate student). Not because we work in places free of noise, because, actually, we’re periodically exposed to rather a lot of construction-type noise at UBC. Not because we don’t do heavy lifting and crouching. Not because we don’t have to walk to work 6 miles through the snow in running shoes, uphill both ways, either.

Much of the data over at Jobs Rated strikes me as completely divorced from reality. Mathematicians, supposedly, work 45 hours per week – in my experience, 50-60 is more realistic. Low stress levels, just because we don’t operate heavy machinery? Who are you kidding?

An annual income of 94K sounds like at least an associate professor at a large research university. Before you get there, you’ll usually have to spend 4-6 years in graduate school (average income at UBC: 20K per year), 2-6 years in one or more postdoc or temporary positions (minimum salary at UBC is 40K; 50K is considered high), and another 4-7 years as tenure-track assistant professor (starting salaries at UBC are around 70K). You’ll also have to make several long-distance moves. That’s the best case scenario as far as the money is concerned. Many more people end up at smaller schools, with significantly lower salaries and higher teaching loads, or as perennial “seasonal” instructors with no job security.

Then there are questions that Jobs Rated did not ask at any particular point: how many of us have a choice of where we want to live if we want to stay in this profession? How many get separated from their spouses or partners for years because they can’t get jobs in the same city or state?

Why do we do this, then, instead of going into real estate or something? Because, first and foremost, we are attracted to mathematics. We enjoy learning mathematics and doing research work. We enjoy working with students – admittedly, not all the time, but nonetheless. We like being able to work mostly on our own schedule, even if the flip side is that we might end up working at home well past midnight. We’re competitive and we appreciate a good challenge, be it mathematical or professional. Curiously, Jobs Rated seems to view competitiveness as only a negative feature and a stress factor, but doesn’t understand that boredom can be stressful, too.

According to the Jobs Rated standards, the best job in the world would involve sitting in an office for several hours a day, not doing anything in particular, and getting paid well for it.

But that’s not what we do.

Blaming the mathematician

Paul Wilmott explains a couple of things about estimating probabilities in quantitative finance:

You are in the audience at a small, intimate theatre, watching a magic show. The magician hands a pack of cards to a random member of the audience, asks him to check that it’s an ordinary pack, and would he please give it a shuffle. The magician turns to another member of the audience and asks her to name a card at random. “Ace of Hearts,” she says. The magician covers his eyes, reaches out to the pack of cards, and after some fumbling around he pulls out a card. The question to you is what is the probability of the card being the Ace of Hearts?

Of course, if a card is chosen at random from an ordinary pack of 52 cards, the probability of it being the Ace of Hearts is 1/52. But is that really correct? What if this is not a math problem, but instead you are indeed watching a real-life magic show in a theatre? Do you really believe that the magician doesn’t know exactly where the Ace of Hearts is? Thus the “real” question is: how likely is it that the magician’s script calls for him to draw the Ace of Hearts? That’s certainly one possibility; but there are others, for instance the magician might pull the card from the pocket of an unsuspecting audience member.

A member of wilmott.com didn’t believe me when I said how many people get stuck on the one in 52 answer, and can’t see the 100% answer, never mind the more interesting answers. He wrote “I can’t believe anyone (who has a masters/phd anyway) would actually say 1/52, and not consider that this is not…a random pick?” So he asked some of his colleagues the question, and his experience was the same as mine. He wrote “Ok I tried this question in the office (a maths postgraduate dept), the first guy took a fair bit of convincing that it wasn’t 1/52 !, then the next person (a hardcore pure mathematician) declared it an un-interesting problem, once he realised that there was essentially a human element to the problem! Maybe you have a point!” Does that not send shivers down your spine, it does mine.

Once you start thinking outside the box of mathematical theories the possibilities are endless. [...] A lot of mathematics is no substitute for a little bit of commonsense and an open mind.

I’ll get around to arguing with Wilmott in a moment, but let me first tell you about the number 52.

Numbers, of course, are abstract concepts. They don’t have to be associated with counting cards, apples, oranges, or anything else. How, exactly, do we define them in the abstract? Here’s how this was explained to me back when I was an undergraduate math student. We start from the Zermelo-Frankel axioms of set theory, and then proceed as follows.

• The Z-F axioms guarantee that the empty set exists. We define 0 to be the cardinality of the empty set.
• Consider the set whose sole element is the empty set. We define 1 to be the cardinality of this set.
• Now consider the set whose elements are the empty set and the set whose sole element is the empty set. The cardinality of the new set is 2.
• Repeat this 50 more times, and you get to 52.

Now, we don’t actually go through this procedure every time we have to use an integer number, let alone fractions. The point is, though, that mathematics deals with idealized abstractions and that we tend to be well aware of our limitations as far as real-life problem-solving is concerned. Ask me to solve the differential equation and I will tell you, with 100% certainty, that . But is this really the equation that you should be trying to solve? That’s where the mathematician needs to hear from someone who actually understands the context. Pure mathematics, alone, cannot speak on that matter.

Wilmott is right to say that it is a problem when mathematics gets to overrule common sense. His diagnosis of the underlying causes, though. gets it exactly backwards. The problem isn’t limited to applications of mathematics, either. Here’s the actor Philip Seymour Hoffman talking about his latest movie Doubt:

“What’s so essential about this movie is our desire to be certain about something and say, This is what I believe is right, wrong, black, white. That’s it. To feel confident that you can wake up and live your day and be proud instead of living in what’s really true, which is the whole mess that the world is. The world is hard, and John is saying that being a human on this earth is a complicated, messy thing.” Hoffman paused again. “And I, personally, am uncomfortable with that messiness, just as I acknowledge its absolute necessity. “

And that’s the real “human element” at work. Uncertainty and doubt have been a part of the human condition from time immemorial, but so has our discomfort with them, our struggle against them. We want security and certainty. We long to be reassured – by religion, medicine, mathematics. We want to be told what the future will bring and we want a 100% refund in the unlikely event that the prediction fails.

There’s a sense of security in having a formula that lets you make predictions. You get to print nice glossy brochures with charts, graphs and tables, citing scientific publications in top journals. The formula, though, is only as good as the assumptions that went into setting it up. If those are true – every single one of them – then the mathematically predicted outcome is inevitable. Just like we would like it to be. We don’t want to read the fine print.

In practice, there’s usually at least one unspoken assumption that fails to hold, namely that the system in question is isolated and there are no more variables to be taken into account. Sometimes it’s reasonable to consider the system as if it were isolated. Other times, it’s not. How do we know? Maybe, really, we don’t.

We don’t like to worry about it, though. We prefer to accept the mathematical solution to the easier version of the problem. I’ve seen it in every calculus class I’ve taught. The “word problems” are grossly simplified versions of real-life situations, simplified so that the problem can be solved using first or second-year calculus. Most of them are worded so as to make it clear which formula should be used, and if that’s not obvious right away, it will be after several repetitions in class and on the homework. When I try to ask the students to consider what unspoken assumptions are being made or in what data range the solution will no longer be correct, it tends to fall on deaf ears, because that’s not a part of the problem, is it? Open-ended problems – find a good approximation for something or other, using your common sense to determine what’s “good” – won’t get me far, either, because how exactly am I going to grade it on a test?

Mathematics is not the enemy of common sense. Intellectual laziness is.

Oh, and those arrogant mathematicians from Wilmott’s story who didn’t “get it”? Sounds too much like a ratemyprofessor complaint. “The first guy” might have been told that this was “a math problem” in a way that suggested strongly that the human factor should be disregarded, as it always is in those calculus problems I’ve just mentioned. The “hardcore pure mathematician” may have been immersed in his work, as we often are, and did not appreciate the interruption. Or he may have suspected one of those “mathematician jokes” that paint the mathematician as a real-life idiot and make the layperson feel oh-so-glad that he’d never learned algebra. Sure, there are mathematicians who live in an ivory tower. There are others who don’t. Consider all scenarios and do not make unwarranted assumptions.

The Monty Hall problem

On a plane flight back from a recent trip, I watched the movie 21. The plot, advertised as “based on a true story”, is roughly as follows. (In case you have not seen the movie and would like to, I will try to avoid major spoilers.)

Ben Campbell, an idealistic and somewhat naive MIT student, impresses a math professor (Kevin Spacey) by answering correctly a couple of tricky questions in class. Soon afterwards, the professor asks Ben to join his card-counting blackjack team in return for a share of the profits. The team travels to Las Vegas on weekends, plays blackjack at major casinos, and wins millions of dollars by placing themselves strategically at the right tables and employing the card-counting techniques taught by the professor. Ben refuses at first (“and if you tell anybody, I’ll make sure that you won’t graduate”), but there’s no other way that he can pay for his dream med school, and he’s attracted to a female student on the team, and besides, if he didn’t join the team, there would be no movie, so guess what happens.

That’s about the first quarter of the movie, and I’ll leave it there, because this is already enough to raise serious questions about just how close to a “true story” we are here.

My first question was, has there really been an MIT math professor who made a fortune off a team of student card players? Wouldn’t that be serious professional misconduct, and would an MIT professor (not a bad job) really take this sort of risk? As it turns out, the movie is somewhat loosely based on the adventures of a real-life MIT card counting team (one of several that MIT has had over the years). However, the teams were all entirely composed of, and run by, the students. There were no professors involved, and the Spacey character is completely fictional.

Which also preempted my follow-up question. There’s a classroom scene where the Spacey character asks his students what applications of Newton’s method they know. A student suggests, “Nonlinear equations?…” Spacey responds along the lines of “Yeah, that’s very clever, because this course is called Nonlinear Equations. Why don’t you tell me something I don’t already know.” My impression was that Spacey’s demeanor, and this exchange in particular, was a little bit too snarky. A professor is not supposed to do that if he (or, especially, she) wants to get good student evaluations.

The user comments on IMDB include several reviews by authors who appear to be well familiar with casinos, blackjack and card counting, and are not entirely happy with the treatment of the subject. I’ve never played blackjack, or been to a casino, but their criticism makes sense to me.

But here’s the reason why I’m writing this post. How exactly does Ben manage to impress Spacey’s character? Spacey asks him the following question:

You’re on a game show. The host asks you to choose one of three doors. Behind one of them there’s a car, behind each of the other two there’s a goat. You pick one door, say Door 1. The host, who knows where the car is and is not allowed to reveal that information, opens Door 3, behind which there is a goat. He then asks you if you want to choose Door 2 instead of 1. Is it to your advantage to switch?

In case you’d like to think about it, the rest of post is behind the cut.