Terraced scans and writing math in TeX


Imagine going to a bookstore and finding an interesting book. How does a person do such a thing? One possibility would be for this person to pick up the first book ey sees, read it cover to cover, and then do the same with the next book, and so on. If you’re like most people, you’ll consider this an absurd algorithm.

This algorithm is an example of a depth-first search. The idea is as follows: go as far as you can along the first path you find; when you can’t go any further, back up until a new unexplored path shows up. Keep going until you’ve explored all possible paths, or at least as many as necessary.

The depth-first search doesn’t work very well when looking at large data sets because it takes a long time to explore each path in its entirety.

A better plan is to look at a lot of data in a cursory manner and try to generate clues about which avenues are worthy of further exploration. Returning to the bookstore example, we might try first looking for a section that interests us. If one wishes to read a popular science book, for example, ey would be advised to start in the popular science section of the bookstore.

So, the first clue would be to start in the right section. But there are probably still many books in that section, far too many to read them all cover to cover, one at a time. So ey would look for further clues. Some of the books have titles that sound intriguing, so ey might take a few of those books off the shelf. Still, there are likely to be too many to read them all cover to cover. To get further clues, ey will probably read a few pages in each of several books to determine which ones have a writing style consistent with eir own tastes. At that point, if ey is determined to buy exactly one, the selection may be somewhat random. But at least ey has narrowed the selection down to a few reasonable possibilities and has selected what is probably a pretty good fit, if not necessarily the optimal one

This process is an example of a parallel terraced scan, as introduced by Douglas Hofstadter and John Rehling for use in artificial intelligence projects. I read about it for the first time in Hofstadter’s essay "The Architecture of Jumbo," which I’ll probably write more about soon. (Interesting piece of trivia: The book containing this essay, Fluid Concepts and Creative Analogies, was the first book sold on Amazon.) The parallel terraced scan is a process that humans find easy and do naturally, without thinking about it. The way it works is just as described above: in the first pass, the scanner (who I will think of as being a person, although it could be a computer as well) very quickly looks through a lot of options and eliminates most of them. On the second pass, the scanner looks at the remaining options a bit more carefully and eliminates most of those. The process continues until the number of options remaining is small enough to give every option a fairly careful analysis. Finally, one very promising option is selected.

 
What does this have to do with writing math papers in TeX? The situation of finding a math paper to read is analogous to finding a book to read. We have various filters that help us decide when a given paper is worthy of consideration, and when it is not. Of course, one filter is the subject area. In my case, I’m significantly more likely to read a paper on number theory than a paper on differential geometry, just because the former is my area, and the latter is not. Another somewhat weak filter is the author. I’m more likely to read a paper written by a well-known mathematician than a paper by someone I’ve never heard of. If one is the author of a paper, and ey wants me to read it, ey has relatively little control over the former, and no control over the latter. So, ey can’t sway me to read eir paper with respect to these filters.
 
One way ey can influence my filter is to change the visual properties of eir paper. If eir work looks like a math paper, I’m more likely to read it than if it doesn’t look like a math paper. And writing in TeX makes it look like a math paper. If ey writes eir math paper in Word, my first reaction is to say "I don’t want to look at this horribly typeset document; looking at it causes me to become angry." And I’ll be unlikely to find out if ey has any beautiful mathematics hidden behind eir hideous typographic choices.
 
I took a course in economic game theory when I was an undergraduate. When discussing signaling games, the professor told us that in college, students typically do not learn the skills needed in their jobs; however, employers tend to be happier to employ economics majors than, say, anthropology majors regardless of the skills they have. The economics majors have signaled that they are willing to work hard on what may be tedious tasks. By contrast, the anthropology majors have pursued a hobby. Of course, I don’t like the implication here: that anthropology is more fun than economics. (Give me the choice between only those two majors, and I wouldn’t have the slightest hesitation jumping to become an econ major.) But the point still stands: sometimes, it’s a good idea to send signals that are likely to appeal to the judges.
 
I have a mental picture of papers floating around in the air, clamoring for my attention. Each paper sends signals in the form of messages of various types about why I should read it. As I alluded to above, these messages are about their content, their likely importance, their author, and their care of presentation. Being written in Word sends a very poor message about care of presentation.
 
And, ultimately, if an author chooses to write math papers in Word or some other non-TeX environment, ey is signaling to me that ey doesn’t care enough about eir work to make it look nice. If ey doesn’t care very much about eir own work, why should I?
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About Simon

Hi. I'm Simon Rubinstein-Salzedo. I'm a mathematics postdoc at Dartmouth College. I'm also a musician; I play piano and cello, and I also sometimes compose music and study musicology. I also like to play chess and write calligraphy. This blog is a catalogue of some of my thoughts. I write them down so that I understand them better. But sometimes other people find them interesting as well, so I happily share them with my small corner of the world.
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