Thanksgiving is Strange

Some years ago, I posted about how Thanksgiving is a difficult holiday during the school year. I just realized how difficult it is for everyone in general.

Presumably, Thanksgiving is a holiday in which families gather together from across the country and celebrate. First, Christmas is just one month after, which seems to be about similar sentiments. These two holidays which revolve around gathering family aren’t too far apart. In fact, I haven’t been home for Thanksgiving for nearly a decade. Christmas (especially since it was during winter break) and New Years were my family gathering times.

In terms of school days off, it seemed that Thanksgiving day and the day after were off. Since Thanksgiving is always on a Thursday, this means that there’s a nice weekend to get back to work. Of course, it’s difficult to travel from work since no time is given off beforehand. This results in nearly empty classes on the Wednesday before Thanksgiving. It seems more reasonable to get Wednesday off. (It’s far less reasonable for a student to take the Tuesday before off in my opinion.)

This is what they (at least in theory) do in Korea with the lunar new year. Three days are (supposed to be) given off: the day before, the day of, and the day after. In China, where the lunar new year (called the Spring Festival, oddly enough) is a huge deal, two weeks are given off (the week before and after). The reasoning I heard about this was that since China is such a large country, people needed more time(!) to travel. And in the Korean version of Thanksgiving too, the same rule applies: the day before, the day of, and the day after are all part of the holiday.

So for a holiday where the point seems to be bringing the family all together, they sure do make it difficult to try to do so.

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Limitations in Fantasy Movies and TV

I have a confession to make: I am not a huge fan of the Lord of the Rings movie trilogy. Yes, I am one of “those” people, who like the books more than the movies (of virtually everything). But at long last, I think I can explain myself. So first, I think that Jackson’s Lord of the Ring trilogy is the best adaptation that could ever have been made. I’m probably exaggerating, but with good purpose.

I think that the major things I love about Tolkien’s book series cannot be adapted to a movie. As I learn a little more about movies and TV, the more I understand that the constrictions that the media have. A good movie tells a tightly told narrative. Meandering is a vice, not a virtue here.

But the story behind Lord of the Rings is, in my opinion, not the main draw of the series. The world of Middle-Earth is. And while the movies provide gorgeous views, real-to-life buildings, and songs in the actual (and appropriate) constructed languages, the mythos and history of the place seemed lacking. Elrond didn’t seem like the wise elf, nostalgic for the millennia gone by, that he did when recounting the elder days. Since the flashbacks were largely treated as exposition, there didn’t seem to be anything legendary or ancient about it, at least not compared to the “current” legendary and ancient times.

All the asides and extraneous bits that made me fall in love with Middle-Earth must be cut from the movies because they don’t fit into the story. Books are free to give a brief aside about people we know nothing about and who make no impact into the greater story. But that can’t happen in a movie. And for good reason, the pace of a movie is forced on you, while you can take as much time as you want when reading.

As another example, I love the A Song of Ice and Fire series by George R. R. Martin, but I’m a bit lukewarm on the HBO Game of Thrones series. (I’ve only watched the first four seasons, so no spoilers please.) Not because of the story, especially since I thought certain elements were done better in the TV show than the books. But because of the immense history and all the people who have to be cut or merged in order to work into a story.For example, how many siblings did Tywin Lannister have? The attentive TV watcher might

For example, how many siblings did Tywin Lannister have? The attentive TV watcher might say: one, Kevan Lannister. But in the books, he had four siblings: one sister and three brothers. I had to look that up, but I knew he had more than the one, because Jaime has an extended conversation with his aunt and Tyrion has a memory about his favorite uncle.  The backstory behind all of the major people in Tywin’s generation has been thought through and you can tell, because some non-essential bits are thrown throughout the book. The War of the Ninepenny Kings, which without looking up I wouldn’t have been able to explain, was mentioned enough that I remember the name and know who fought in it.

So all this to say, it seems that being a “book” person is not just some pretentious conceit, but may be part of the media itself.

Functors Between Metric-Enriched Categories: Is This A Stupid Idea?

Note: I’m hoping to get some comments about this, so if you have some thoughts, please leave a comment.

Let Met denote the category whose objects consist of metric spaces (for convenience, we will allow metrics to take the value of infinity) and whose morphisms are (weakly) contractive maps (aka short maps, nonexpansive maps, 1-Lipschitz maps), or more precisely functions satisfying d(f(x),f(y)) \leq d(x,y)Met is a monoidal category where X\otimes Y is the product space with metric given by d((x_1,y_1),(x_2,y_2) = d_X(x_1, x_2) + d(y_1, y_2) (the identity object is the singleton).

It is my (perhaps mistaken) opinion that a lot of functional analysis can be done using categories enriched in Met. (Actually, I think that for more generality, one ought to replace metric spaces with semi-metric spaces (aka quasi-metric spaces) where distinct elements can have zero distance. It may also be possible that Lawvere metric spaces are the appropriate choice here, but I’m not yet convinced of this.) Of course, one “trivial” yet important example is using the discrete metric. So for any two objects A, B in a locally small category, we can define a metric on \operatorname{Hom}(A,B) by $d(f,g) = 1$ if $f\neq g$ and $d(f,g) = 0$ if $f = g$.

A category enriched in Met enables one to talk about approximately commuting diagrams. This has been explored in approximate Fraïssé limits, though I don’t know enough logic to understand it.

Given two categories \mathcal{C}, \mathcal{D} enriched in Met, we can define a type of continuity (except that word’s already taken) where a functor F: \mathcal{C} \to \mathcal{D} is “continuous” if for any \epsilon > 0, there exists \delta > 0 such that for any objects A,B\in \mathcal{C} and any morphisms f,g\in \operatorname{Hom}(A,B), if d(f,g) < \delta, then d(F(f),F(g)) < \epsilon.

Apologies for the underdeveloped ideas, I have been thinking about this for quite a while (throughout my time as a graduate student) and I have some trouble formulating what I want to say. Talking to a few people about this did not generate a terrible amount of interest, but I was curious if people had any insight as to:

  1. Has there been any work done along these lines?
  2. Do you think this might be a potentially interesting idea?

My category theory is limited, and what I’ve learned does not seem to have this type of idea in mind, but I’d love to hear some thoughts!

AI and its effect on the aesthetic of gameplay

It is a well-known fact that nowadays no human chess player can beat a computer AI. Relatively recently, Lee Sedol, one of the best Go players, lost 4-1 to Google’s AI AlphaGo. The reason I bring up this old news is FiveThirtyEight’s recent article “Are Computers Draining The Beauty Out Of Chess?” Like the aesthetics of mathematics, I find it difficult to explain what aesthetic sense a game of chess or Go might have, and since I’m not a great chess or Go player, it’s likely that I have a limited sense of it anyway. Much of it, in my limited experience, is a mixture of naturalness and surprise. A beautiful move is both surprising, and yet in hindsight, appears to be the most natural thing in the world.

I think back to reading Yasunari Kawabata’s The Master of Go. In Japan back then, there were strong traditions and rituals involved in playing the game, to the point where victory seemed secondary to the play of the game itself. Just as the loss of the game in that book may be symbolic of the waning traditional values in Japan at the time, I feel that Lee Sedol’s loss against AlphaGo may portent to a loss of aesthetic value in the game.  Unlike most art, the primary focus of games is to win. And to win now means to behave like a computer. Where playing like a human resulted in beautiful moves, playing like a computer seems to result in safe, calculated movement.

Honestly, I’m out of my element here, talking about many subjects that I’m not an expert in.

The Future of this Blog

Casey Neistat ended his vlog today. I’m a fan of his work, and while I’ll miss the daily vlogs, I look forward to whatever new projects he puts out. But what interested me the most was his reasons. Contrary to the thought that vlogging every day would be difficult, it seems that rather than encouraging innovation, the vlogs became habitual.

I couldn’t help but make comparisons with my blog and a similar project. Of course, it’s presumptuous to compare my tiny blog to a YouTube channel with more than 8 million subscribers. But this too was a project to force myself to be more creative and to actually produce something (maybe not stunning videos, but something). And the small scope and audience of this blog make me wonder what I’m trying to accomplish with this blog. Of course, the size of my audience doesn’t matter if my purpose doesn’t require a large audience. But my purpose will reflect my hopes and goals for this blog.

I’m afraid I don’t have any answers to this question. While I won’t stop blogging now, I find myself needing to reflect on what the future holds for this blog, and if the lack of purpose is the reason for the previous lack of motivation to post.

Some Thoughts on “A Breakthrough in Higher Dimensional Spheres”

Since I have an interest in the popularization of math, I was excited to hear about a new PBS series called “Infinite Series” and their first YouTube video “A Breakthrough in Higher Dimensional Spheres.” So I decided to write down some thoughts I’ve had about this video and some general trends in pop math.

My general impression of this video is that it’s good. It strikes a good balance between being correct and understandable for the layperson. Many articles about math tend to fail at this, using inappropriate analogies to try to illustrate the point. I am slightly disappointed that some details weren’t provided (more on this later), but it’s extremely hard to balance the level of detail with being entertaining.

To get to the details, the video starts with an introduction the show and its premise, which seems to be current progress in mathematics. It’s not a bad idea, but since most mathematical ideas are built from older ones, this will either limit the topics covered or require each a portion of each episode to go over some background material. Based on how this episode went, I’m going to guess both factors will come into play. I don’t expect to see an episode on the Kadison-Singer problem or the classification of nuclear C*-algebras, for example.

The introduction of spheres and Euclidean space of arbitrary dimensions was good. The sphere packing problem was stated well, but I thought the examples and explanations lacked some important details. The 2d sphere packing is based on square grids and hexagonal grids. Even outlining this fact in the image would have gone a long way to illustrating how that works. It would also help in understanding the higher dimensional generalizations. The 3d case also does little to explain how the sphere packing works and the shapes involved. Also, it’s unclear if, as the picture seems to indicate, the 8d case is a “duplication” of the hexagonal sphere packing. Also when discussing the general case, I think it would be helpful to know where the difficulty lies. Is it the case that we have candidates for other dimensions, but can’t prove that it’s the best as it was for the 3d case? I felt that the sphere packing problem and its solutions were inadequately explored.

The counterintuitive nature of higher dimensional spheres, on the other hand, I thought was explained well. The ratio of the volume of the sphere to the volume of the circumscribing cube going to zero was interesting. (This might be an interesting worksheet in a calculus class, I should make a note of it.) The analogy with the basketball court and the grain of sand was good, due to the fact that such an object with all those properties is impossible to conceive of in our limited space.

Despite some flaws, I think the video was pretty good. Its discussion of the sphere packing problem itself, I think was of moderate success. But the discussion of higher-dimensional spheres worked well to highlight its counterintuitiveness. I am looking forward to seeing more videos from this program, and I hope you’ll join me.

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.