No Classics in Mathematics?

As far as I can remember, there has been a collection of the Great Books of the Western World in my home (though to be fair, my memory does not go as far as many other people). Looking at the collection, what strikes me most is how few mathematical texts there are. In total, there are four authors writing about mathematics: Euclid, Archimedes, Apollonius, and Nicomachus. They are confined to a single book, and all of them are from the ancient world. To be fair, part of this is how I classify things; some people might classify the works in this collection by Newton and Fourier as mathematics, but I consider them part of physics. And of course, the distinction between subjects was not clear then. (Hence the Augustinian quote: “The good Christian should beware the mathematician and all those who make empty prophecies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of hell.” Here mathematicians, astronomers, and astrologers are all grouped together.) Science and mathematics (and philosophy) were all studied together and mixed together. Though some people, such as Descartes and Pascal, made mathematical contributions, but only their philosophical works are included.

In general, I wouldn’t be surprised if most well-educated non-mathematicians could only name at most one mathematical text (namely, Euclid’s Elements). In fact, most mathematicians (myself included) rarely read any historical mathematical texts. I can think of a few reasons for this.

One, the language, especially in older texts, tends to be archaic in unexpected ways. This is especially problematic for non-mathematicians, since reading math is already similar to reading in a foreign language. The vocabulary hasn’t quite settled and so strange words are used to describe familiar concepts, which makes reading difficult. The notation is often quite different from what we’re used to. Using letters to denote variables didn’t happen until the end of the 16th century with Viete. And it was until Euler that we had much of the modern notation that we use today. Though often times, the notation of older texts is “translated” into the mathematical notation that we’re used to today (in addition to actually being translated from their original language).

Two, the original conception of an idea may be notoriously dense and opaque, but the efforts of later people can clarify that idea and make it much easier to understand. In that case, we give credit to the original, but then never read it again. The work of Galois is notoriously difficult to read, but we name the theory after him since he first conceived many of those ideas. This is not to dismiss or minimize these people’s works. Writing new ideas without a language to describe them is going to be a difficult task, and undergoing such a task should be praised. (Though, disgesting and presenting in a clear manner these ideas is also an important task, though it seems far less glamorous.) So the original text are often abandoned for clearer exposition written by people who worked hard to digest the original work. Then this is iterated again and again until we have our modern, super clean presentation of classic works.

All this to say, while I understand it to an extent, I find the current state as unfortunate. As a mathematician, I should read the classics to better understand the material and motivation behind several turns in thought. As a person interested in mathematical popularization, I find it problematic that rather than turning to some great works of the past, we often resort to presenting a pre-digested, cleaned-up version of a classic. Though, it’s hard for me to go tell people to read the classics, since I have not done so. Maybe I’ll try to remedy that.

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Thoughts on the Game Mini Metro

Among a subset of mathematicians, there seems to be a fascination with metro systems and their maps. Part of their appeal is how well they seem to resemble the mathematical mindset from taking the concrete to the abstract. The system is boiled down to several dots connected by colored lines. And despite its seeming simplicity, the resulting diagrams are often quite complex and has a sort of beauty to them. I’m sure there’s quite a lot to be told about a city based on its metro system, but I wouldn’t be qualified to say.

I bought the game Mini Metro while it was on sale on Humble Bundle and have been playing it quite a bit for the past few days. It is a game where you build metro lines and try to serve as many passengers as possible before a station becomes overcrowded. The minimalistic art style really suits the gameplay, since it feels as though one is playing on a metro map. The use of the Helvetica font sells the concept further. The gameplay is easy to pick up, and trying out strategies is a nice little exercise for the brain. Though I thought this game would suit playing on a phone, I find my fingers made the controls feel a little clumsy. I suppose if I had a stylus, this would be less of a problem.

It seems that minimalism can be too abstract sometimes for some people, but what makes this game so amazing is how despite its minimalism, the apparent symbolism becomes quite immersive and the sense that basic shapes can represent commuting people and trains so immediately in one’s mind.

A Few Antedotes about Time

The clocks in my house run five minutes fast, and it continually reminds me of a story that happened this summer in China. A friend of mine, who apparently constantly runs late, had the ingenious idea of setting his watch ten minutes fast, so that when he’ll be ten minutes earlier than he would have been to whatever meeting he needed to attend. This backfired when he was giving a talk. Thinking he had ten more minutes, he continued speaking for five more minutes and delightfully announced that his talk would end early. Of course, in reality, he went five minutes over.

While it’s generally better to end early than end late, I generally want to act out the German stereotype of being precisely on time and end my talks at the appointed time. One year, when I taught classes, I would give ten minute warm-up exercises for the students to do before lecturing. Whenever I gave talks that year, I would end ten minutes early. It was quite troubling; apparently, I had gotten so accustomed to lecturing for 40 minutes that my internal clock prepared for exactly 40 minutes worth of material for my talks as well.

Some people are surprised that I wear a watch (especially before the advent of smartwatches), but it’s a much more innocuous way of telling the time. As I mentioned before, I hate going over time, especially when teaching (no ending bells in the University of Oregon). So early on, before I got accustomed to lecture timing, I would constantly check the time on my phone to make sure I wouldn’t go over. A student noticed this and angrily commented on my teaching evaluations that I was constantly checking my phone.

For those living in the US (and not in Arizona or any place without daylight savings), don’t forget to set back your clock tonight (if you are old-fashioned like I am).

Laconic Proofs: Can Mathematics be Poetry?

Mathematical writing is a narrative form. The development of definitions and theorems are not unlike the story of growth and struggle of a character. A well-written math paper or book has a narrative structure with an initial goal as the introduction, some definitions and lemmas as the rising action, a major theorem or corollary as the climax, and a resolution, which can be surprisingly diverse. This is a topic that I would like to explore in the future (provided I can be coherent enough to talk about it).

But what can we make of the short proofs found in the literature? (No, not “left as an exercise to the reader”.) I mean what do we make of Littlewood’s hypothetical shortest dissertation (pdf)? There is a game of sorts to find who can publish the shortest paper. But beyond ego boosting, there’s a certain aesthetic to these types of laconic proofs that demands extra attention from its reader. It reminds me of laconic poetry that I once read, and similar demands that it made. Now, I confess that very few mathematicians bother writing in verse, but there’s a certain parallel between these two forms. Their brevity betrays their depth. It takes so little time to read, but so much time to unpack. Not all of them have this quality, I would say the counterexample to Euler’s conjecture (pdf) lacks any depth. But a one page proof of the irrationality of pi or a one sentence proof that every prime of the form 4k + 1 can be written as the sum of two squares has a charm and beauty of its own that I dare to say is poetry.

Blogging every day in November

In the first term of my undergraduate schooling, I took a math course that was considered one of the most difficult courses in the university. Given that, ultimately, I would go on to obtain a doctorate in mathematics, one could say that that decision changed my life. It was that challenge that, among other things, appealed to me. The course was quite difficult for me, but I wanted to be challenged and pushed to work harder than I ever did before. Throughout my life, whenever I looked back to the past, I always regretted not having worked harder and accomplished more. I didn’t really give myself enough credit; I did work hard, at least at times. It’s easy to regret the past when the troubles and temptations of the times are easily forgotten or diminished. But what became clearer over time is that the difficulty in working hard is mostly in the motivation. It is, in a sense, easy to work hard when faced with an explicit, external challenge (at least for me). The work needed to pass a difficult course is more tangible and immediate than most work I have nowadays, which is often vague with unremarkable incremental progress.

Of course, one solution to this is to make explicit the vague challenges that I want to face. When I taught, I would have a difficult time writing lecture notes for myself despite being a useful tool to have. It became much easier to write them when I posted them online. Writing a set of lecture notes became a much clearer task when I had an actual audience to write for. Also, it was much easier to write lecture notes when I had students emailing me asking when the next set will be available.

Some years ago, I decided to try to write every day in the month of November. After some thought, I have decided to try that again this year. While there’s not likely to be a demand for my silly diary entries or half-baked mathematical ideas, making a public announcement is likely to push me to write every day. It might not be the best motivation, especially since the last time I did this, I stopped writing on Thanksgiving.

So on the off chance that someone is reading this, thank you. I hope at least a few entries in this project are of some value to you.

Selective Rigor

Recently, I’ve been informed that very little is known about Pythagoras and in particular, it wasn’t known whether he even proved the theorem of his name and most of what’s known about him is myth. That got me to thinking how often I heard stories, namely about the death of the man who proved the irrationality of the square root of 2, and how I never really checked sources on whether this is true. I also recall (but can’t find) a portrait purported to be of a well-known mathematician (for some reason, I thought it was Lambert), turning out to be a portrait of someone else entirely.

Back in my undergraduate years, when I was first learning about mathematical rigor, I became a little zealous with the idea of rigorous thought. The unambiguity was so refreshing and beautiful that I felt it ought to be carried throughout life. To be precise and careful with everything one says seems like a great aspiration. Of course what actually ends up happening is I rarely talked and when I did talk, I would add conditional statements to everything and depend on exact wording that most people don’t hear. It’s been surprising frequent that I would make a conditional statement and people would hear only the conclusion. It also led to a skepticism and doubt regarding my own opinions. But despite the detriment, I feel it’s a net positive, since it taught me to try to hold my own opinions at a distance and that I can’t ultimately separate myself from my bias, but that if I try, I can go a long way to gaining just a little bit of objectivity.

Before I got to graduate school, I felt that this was an aspiration of all mathematicians. So I was quite surprised to realize that this wasn’t the case. Most people happily deal with imprecision and ambiguity. In some cases, I felt their mathematics also suffered from sloppiness. But in other cases, it seemed as though, despite an ability to be mathematically precise, there wasn’t any desire to extend this to other aspects of life and discussion. People seemed to hold strongly opinions, for example about politics, that seemed hardly justified. If they were doing math, I would think to myself, would they be so bold to make such unjustified claims?

I feel quite happy with my strange aspirations of precision and rigor, despite giving up on being able to speak in perfectly precise language. The more encounters I’ve had with people who don’t have this strange habit, including other mathematicians, the more I think of this as just another one of my own biases that I need to remember to take into account when trying to understand the world at large. And I need to remember that as a vocation, mathematicians can be very precise, but that need not extend to their personal lives.

How I Chose My Major (But Not Really)

This post was initially going to be about the fact that I considered myself more of a philosophically-minded person than scientifically-minded one and how odd I found that when most people associate math with science. But the more I delve into my memories, the harder it is for me to justify such a narrative. So in contrast we have a different story emerging, one where I didn’t have some clear idea of who I was and what I was going to do or try to do. So instead I want to look back at high school, while I was deciding with which major to apply to college, and having narrowed things down to five choices: philosophy, political science, economics, physics and mathematics.

With mathematics, I thought I was kind of good at it. I actually didn’t know what actual math was and when I took my first math class in undergrad, it was eye-opening. But it’s worth noting that the math I had in high school wasn’t terribly interesting to me with the exception of some geometry.

Physics was a little bit more fun, but it was more that the computations made sense. It had a bit of fun experiments, but I feel awkward around making measurements and actually doing stuff.

The philosophical side was formally underdeveloped. I had read some Plato when I was young, but that was about it. Even now I feel underread in philosophy, well mostly because I am. But it is important to note that what compelled me the most about this subject was the awe-inspiring basis in deductive reasoning. It’s difficult to discuss let alone defend one’s own opinions and intuition. But you knew where things stood with reason and it gave grounds for thinking the way I did. It had a confidence building effect. I wanted to develop something in which I felt had a kernel of ability.

I had an interest in political science, mostly in international affairs. I had diplomatic aspirations, which in retrospect seems like a bad idea. The manner in which political structures function is of academic interest to me now in a limited way, but I actually had political aspirations at one point and now this seems completely foreign. I can’t quite pinpoint why I thought this was a possibility.

I took a course in economics and thought it was great. There was something compelling about having formulas and numbers involved in the whole endeavor. I thought this combined subjects in which I had a little bit of talent and a subject in which I had personal interest. I had wanted to know how the economy (and more generally society) functioned.

What compelled me the most was that I imagined this whole thing on a line with social science on one end and the natural sciences on the other (and math was on the far extreme of this end) and here in the middle, as a natural compromise, was economics.

So that’s how I came to choose economics as my major. But then I actually went to college.