Mathematicians say the darndest things!

Dear readers,

If you have mathematically inclined friends or are mathematically inclined yourself, you may know that mathematicians tend to use a very specialized language when discussing their work. This is not a matter of trying to make the subject of discussion appear inaccessible to outsiders (though sometimes there is an element of that). Instead, it’s a communication necessity: everyday language is even more inadequate for discussing mathematics than it is for discussing topics such as music, philosophy or law.

But the terms used in mathematics are rarely invented from scratch; a great number of them are in fact taken from everyday language, but given different meanings. This is why you might hear words like filter (in lattice theory), flag (in linear algebra), ring (in abstract algebra), saddle (in calculus and analysis), sheaf (in topology), soul (in geometry), at a mathematics lecture. Their meanings, though usually inspired by natural language, frequently drift to the point of becoming unrecognizable.

Fortunately, this exchange between mathematics and natural language is a two-way street, and today I’m going to discuss some of the words that mathematicians often use in day-to-day conversations that are derived from mathematical terms. Using them might make you sound strange, but it might also help you express things concisely. I present to you the top 5 words I’ve heard used in conversations between mathematicians (and have been guilty of using myself in conversations with non-mathematicians on occasion). Whether you decide to adopt them or not, you’ll find it illuminating to know their meanings.

Asymptotically
Mathematical meaning: as the independent variable becomes arbitrarily large.
Colloquial meaning: in the long run; looking at the big picture.
Example 1: Since he’s not staying in academia, having his name on papers is asymptotically irrelevant.
Example 2: They do have a significant age difference, but asymptotically it won’t matter all that much.

Corollary
Mathematical meaning: an easily derived logical consequence of a proven result or theorem.
Colloquial meaning: something that follows automatically; something you get for free.
Example 1: If she gets invited to this wedding, her significant other’s presence will be a corollary.
Example 2: I just bought myself a new bicycle, and this water bottle came as a corollary.

Inflection point
Mathematical meaning: the point on a curve at which the second derivative changes sign.
Colloquial meaning: a situation where each new unit of effort affects the result less than the one before.
Example 1: I could have stayed up all night to finish it, but I don’t like to work past my inflection point.
Example 2: We’ve been working for ten hours and the inflection point is getting close; let’s call it a day.

Modulo
Mathematical meaning: the remainder of the division of a given quantity by another (fixed) quantity.
Colloquial meaning: minor details that must be dealt with as a condition for getting a desired outcome.
Example 1: I will definitely join you at the basketball game tonight modulo finishing this assignment.
Example 2: I got the job modulo the background check and some annoying paperwork.

Orthogonal
Mathematical meaning: perpendicular; in probability theory, uncorrelated (refers to random variables).
Colloquial meaning: unrelated; independent.
Example 1: Your observation is very interesting, but it’s orthogonal to the question we are discussing.
Example 2: How important a task is is often orthogonal to how urgent it appears to be.

Are there any other mathematical terms you like to use in everyday life? Share them in the comments!

My first date and stereotypes about mathematicians

Dear readers,

My first date (at the end of high school) involved going to see “A Beautiful Mind” with a girl I liked. At some point during the movie, she told me: “Leonid, please don’t become like this”. While this was a very thoughtful comment, it made me realize that even intelligent and well-educated people may believe that mathematicians resemble John Nash as he is represented in the movie. Another widespread image of a mathematician is that of Grigori Perelman, of the Poincare conjecture fame. This is why this blog post discusses some stereotypes about mathematicians – and their work – prevalent in our society.

Of course, almost all stereotypes (except for the very outlandish ones) are based in reality. The problem with stereotypes is not that they are false, but that they are too general, and substitute a generalization that frequently fails to apply for our ability to judge people for who they are. Stereotypes provide significant savings of mental and emotional effort, which is why they persist, but they also come with a high opportunity cost – that of dismissing a chance to relate to an individual. One goal of my blog is to prevent you from paying this cost when it comes to mathematicians; another goal is to show that they are generally interesting people worth getting to know, which I plan to do through my interview series.

So first, let’s close our eyes and imagine a mathematician based on what we know from popular culture and the media, conjuring up as precise an image as we can, including physical appearance as well as character traits. Chances are, the image you have will look something like this: a white male in his late thirties, with clothes covered in chalk dust and hair in disarray, self-absorbed but with eyes lighting up when working on a challenging mathematical problem. This might be someone who enjoys being alone more than interacting with other people, possesses limited social skills and lives in his own little world.

Now, let’s examine each of these characteristics in turn to see whether they correspond to reality. Rather than rely on data, which is rather sparse, I will draw on my personal experience to give you a more nuanced picture. Along the way we will discover some of the real quirks of real mathematicians and see which characteristics are really helpful to a successful mathematical career and which are a distraction.

We often hear that “mathematics is a young man’s game”, a statement I used to believe, which made me sad to turn 21, an age at which Gauss had a PhD and Galois, eponym of the famous theory, was already dead. In reality, mathematicians mature at different rates, and many achieve their best results after 40 – Andrew Wiles’ proof of Fermat’s Last Theorem, which had remained unproven for over 350 years, is a case in point. On the other hand, mathematics research gets harder with age, though many, like Harold Coxeter, last century’s greatest geometer, continue doing it into their 90s. One of my mentors who has recently turned 67 once told me that he’d like to live forever as there is always more mathematics to do.

Although our statement probably uses “man” generically, we need to acknowledge that women are still much less likely to do mathematics than men professionally. I’m convinced that the reasons for this are social rather than genetic, despite Larry Summers’ infamous claims to the contrary. In fact, several of my best mentors, including my PhD advisor, were women. Yet the question of how best to encourage young women to develop their mathematical ability is challenging. A campaign like Science: It’s a Girl Thing strikes me as the wrong way to approach this problem. On the other hand, Danica McKellar’s books, which encourage girls to excel in high school mathematics, made a favorable impression on me.

I’m equally convinced that social rather than genetic reasons are responsible for Caucasians currently being over-represented among mathematicians. Mathematics traces its origins to Africa, owes a great many discoveries to Indian and Arab mathematicians during the Middle Ages, benefits from important contributions coming out of China, and only became predominantly European in the last few centuries. Even Andrew Wiles’ breakthrough built on the work of the Japanese mathematicians Iwasawa, Shimura and Taniyama, to give a modern example. African American, Chinese, Indian, Korean, Latin American, Persian and Vietnamese students were integral to my MIT Math PhD program and greatly enriched it.

When it comes to sartorial choices, mathematicians I know do have a propensity for prefering informal clothes, because they value function over form. This year I attended several professional conferences, including the largest annual mathematics convention held in Boston in January. Sure, it was no business meeting or fashion show, but none of its seven thousand attendees stood out for their lack of attention to self-presentation. As for chalk dust, it can be seen on those mathematicians who teach old style, and chalk allergy is an occupational hazard of mathematics if there ever was one. Then again, most people from my generation use slideshow presentations to teach and whiteboards with markers to do research.

As for personality, doing mathematics requires an ability to reason about abstract concepts for extended periods of time. This demands commitment and hard work, not unlike making art or playing music. No wonder that the annual Math Department Music Recital at MIT is never short of performers, or that my best friend from the Math PhD program is also a talented artist. Most mathematicians I know, including myself, are indeed introverted and prefer to work alone. At the same time, the majority of mathematical papers coming out today have multiple authors; none of my 8 papers published so far was written solo. So, at the end of the day, mathematics, just like music or art, is meant to be shared with the community.

Now, in many of our minds, being introverted is frequently associated with having limited social skills. I prefer to think of it as a low tolerance for small talk and a limited interest in other people unless they are particularly engaging. For many introverts, myself included, developing these skills is only possible through a concerted effort, and there are many among us who never make the decision to invest time in it. This may make it harder to get to know mathematicians on a personal level, especially for extroverts. However, the rewards of trying to do so may be significant; mathematicians’ opinions about everything from the history of our civilization to human reproductive biology will be sure to challenge your views.

As for living in their own world, the specialized language mathematicians use (the topic of my next post) might give this impression. But most mathematicians I know do actively engage with the world around them, and only a minority tends to dismiss it as a distraction to their work, like the quirky Paul Erdos. Thus, one of my fellow PhD students at MIT has a side career in ballroom dancing, another is a competitive programmer, and still another is contemplating a career in politics. I’ve been very actively engaged in social justice activism until recently, and only my new line of research in infectious diseases made me step back from it. In summary, we live both in our own world as well as the world around us.

Well, I hope I’ve been able to show you that our stereotypes about mathematicians are not all justified, though many are grounded in reality. The one thing mathematicians are usually not, however, is boring. Oh, and that date I mentioned at the beginning? She actually became my first girlfriend. It might be a lucky coincidence that I actually shared a lot of these stereotypes at the time, since I might have reacted quite differently to her comment otherwise. Then again, if I had already been so enlightened back then, we would probably have chosen to see a different movie, like “Rites of Love and Math” – just kidding!