In this post written on a rainy Sunday, I gather the concepts of mathematical induction and distant reading around character network analysis. Probably that it should have been divided into to smaller posts, but I found interesting to discuss in parallel scalability in mathematics and literary studies.
As a mathematician, there are many concepts I discovered in my studies or later that suddenly blew the view I had on the world. During my first year in gymnase (college), I didn’t get a single thing of what was happening at the physics course, where we were studying mechanics, i.e. an application of calculus, without having ever studied calculus before. But when we studied that subject in the mathematics course one year later, suddenly everything became clear: the derivative describes what is happening at a single moment, and by extension at all the others. It is the tendency of the quantity described by a function to grow, remain constant, or decrease. Then the derivative of the derivative… describes how the derivative behaves. Yep, you can do that infinitely. In the physics course, we had studied how to represent the position of a point in space (e.g. a car) at any moment in time by a formula. Deriving it would give us the speed of the car. Speed can be mentally computed by dividing the distance from point A to point B by the time it takes. But more importantly, the speed given by the derivative is the speed at one exact moment in time: it is what happens if A and B are so close that they get mixed up as the same point. Doing the mental computation here is problematic since you need to divide zero by something that looks like zero too. The derivative lets you know the speed of the car like if you were reading it on the dashboard.
I want to draw a parallel between one of these experiences and what I’m working on now, but I feel the need to keep sidetracking.
Unlike calculus, linear algebra was in majority made of pain in the ass pen and paper calculations, but after a few weeks you would discover that you had dived in spaces made of so many dimensions your brain had difficulties to see how to move through this. At first you were studying a space of 2 dimensions, then a space of 3 dimensions… only to realise that not so many things had changed. And then a space of 4 dimensions. (You see the pattern.) You can do whatever you want, you have received the means to find your way there through calculation, and maybe later you’ll “see” something (and you’ll help me find a nice and clean answer to this question people have asked me so many times: “Can you see in four dimensions?”). Ever done one of these magical PCA’s but never understood how it works? That’s linear algebra, and it’s never too late.
There are many ways to define a straight line: one is to pick a vector, choose a point and that’s it. How does that work? Grab a sheet of paper: a vector is defined by a direction (choose a point anywhere on the physical border of the page), an orientation (towards that point or in the opposite direction) and a length. At this point of the exercise, you have drawn a nice arrow. This arrow is a vector, and you can put it anywhere you want on the page: it is the same vector if it has remained parallel to the original, has the same length and still goes with the same orientation. You do not need to fix it to a point. An infinite number of lines are defined by being perpendicular to this vector, but a unique straight line is defined by being perpendicular to the vector and passing through this point. That’s it for the two-dimensional case.
Zero, One, Infinity
Euclide defined geometry with five axioms from which all the rest could be deducted. Four of them are indisputable, but the fifth–stating that given a straight line and a point not belonging to that line, then there exists one and only one straight line parallel to the first line and passing through this point–lead to an open discussion: is it really an axiom or can we deduce it from the four others? Adopting an opposite approach to that question, in the 19th Century some mathematicians began to ask: what if we keep the first four axioms and we drop or replace the fifth one? What if, given a straight line and a point, we could draw more than one parallel? Or could not draw any? I studied the consequences of these questions during the first semester of the geometry course: stating that there are no parallels leads to spherical geometry (among others) while stating that there is more than one (meaning an infinite number) leads to hyperbolic geometry, which incidentally was the subject of my master thesis. This pattern has been stuck in my mind since: 0, 1, infinite. In my master thesis, I discussed one of the models used to represent hyperbolic geometry in two dimensions: the Poincaré half-plane model.
In this model, you draw a straight line and consider only one of the two halves that it defines. The straight line is the infinite, just like the farthest point at the end of any straight lines you would draw perpendicular to the “infinity” line. Now we define what a straight line is in this model, with this geometry, because it is not the straight line we know anymore. In the real world, the one from our college years, a straight line is often defined as the shortest way from one given point to another given point, extended infinitely in both directions. Here, this is more or less the same, but the metrics–the way to measure the distance between two points–has changed. For the sake of coherence, we are renaming the “straight line” by its mathematical name: the “geodesic”. In that model, given two points, there are two possibilities of geodesics (in fact one, but visually two), a fact that I am not proving here. I need you to believe me: a geodesic between two points is the arc delimited by these points that is part of the half-circle passing through these points and whose center is on the “infinity” line. The (visual) second type of geodesics occurs when the two points are aligned on a Euclidean perpendicular to the “infinity” straight line: in this case the geodesic passing through these two points is the perpendicular itself. Now what about parallelism? Well, given a geodesic and a point that does not belong to this geodesic, a parallel to the geodesic passing through the given point is any geodesic that does not cross it: draw any half-circle with the center on the “infinity” line, containing this point and not crossing the geodesic and that’s it: you have your parallel. Thus now you can see that there is an infinite number of possibilities.
You often find this pattern in mathematics, since zero is the neutral element for addition, one the neutral element for multiplication, and the infinite the place where all the stuff you don’t control disappear. Most of the time, this can be linked to mathematical induction: given a statement you want to prove, zero is a special case, one is a starting case that is easy to check. Then, if you can prove that when assuming the statement is true for any natural number n it implies that it works for the number n+1, you have proved it for any case up to the infinity.
One, Two, Infinity
We studied statistical hypothesis tests during the whole second semester. The tests were classified along two dimensions. The first one was the type of variable: nominal, ordinal or numeric. The second one concerned the number of samples: one, two, or more. (For the sake of precision, there was also a special case: one sample but two measures instead of one.) The course proposed a statistical test for each combination and the whole thing was presented in a nice table that made their exam revisions easier. Here, the connection I am trying to infer may be perceived as exaggerated, but I cannot help thinking that all this is related, taking the risk of finishing this post on an open ending.
Nowadays, I am exploring the relatively recent concept of character network analysis, that is the study of the characters of a novel, with a focus on their relations. Most of the time, a character network is the model of a novel’s discourse: it positions characters one to another based on their interactions in the text. There is a lot to ask and answer on what this object means, what it represents relative to the character-system, to the discourse and to the story. In my works, I focus more on network analysis and statistical methods, and this is were all this mathematical preamble links in my opinion to distant reading (but I do not know if this helps somehow). Here is the approach I have started following when working on character network analysis methodology.
Then comes the facultative question of how to visually represent it, which is easily solved in the case of a corpus of one.
Two Works (Comparison)
I should not forget to mention that at this level of observation it becomes more than necessary to have an automatic method in order to build the character networks. In my case I use back-of-the-book indexes of characters: it has the advantage of providing a disambiguated table of occurrences but the disadvantage of making scalability nearly impossible (at least with manual indexes).
Many Many Works (Massiveness)
As I feared, I haven’t solved what is a mystery for me. In fact I haven’t tried very hard. Is the “one, two, infinite” rule the distant reading equivalent of the “zero, one, infinite” rule in hard sciences? The same pattern or a sibling one? Does it allow a transfer of methods? Am I right to care about that or just a bit too obsessed by patterns in science? Maybe we should we call it distant reading induction, but I am quite convinced that this is redundant by definition.