The Elements of Style

What can a small book for fiction writers offer to a mathematician? A lot, it seems. The Elements of Style, by William Strunk and E.B. White, is the most important book I’ve read this year, and reading it was an absolute joy. At just over 100 pages, it takes little time, but demands great attention. With watertight prose and unyielding attention to detail, the authors marshal the reader’s verbal reasoning into order. When I finished, my mind felt like a newly-neatened room. 

Anyone who writes, or indeed speaks, will find the book useful. It explains how to construct phrases that are faithful to one's own thoughts, and that sympathise with the reader’s difficult task of interpretation. But the book is especially important for mathematicians, because it reveals how reason is most properly expressed: with order, completeness, and attention to detail. The book is mathematically beautiful, in that it both emphasises and demonstrates elegance and parsimony. It shows how constructing a sentence is like constructing a proof: there are many wrong ways to do it, a few right ways, and perhaps only one way to make it sing, where its structure reflects its content, and its construction places the emphasis in all the right places. The ‘singing' quality isn't the product of a gimmick; in fact, the discerning writer won't strive after it at all. Rather, it's accidental to the writing process, or rather, the cutting process. White tells the writer to chop away anything superfluous, leaving only a solid gem of composition. That is where style shines forth - not in the trappings, but in the substance. The author cannot help it.

This stripping-away is easier for the mathematician than for the prose writer, due to the different structures of their two languages. Mathematical expression is stingy by construction. It rarely admits the ornamental. As a result, when writing proofs, pure style often comes through. Still, mathematicians can get trapped by an over-emaphasis on formality. At best, this makes for a proof that’s boring to read, but at worst, it can impede reason. This is the second key virtue of White’s take on style: not only does good style aid the reader, but, if the voice on the page squares with the voice in one's head, writing becomes an act of open communication with oneself. That’s where creativity happens, where shoddy arguments are shown for what they are, and where strokes of brilliance can be captured, held, and processed in a way that wouldn’t be possible with unrecorded thought. 

The final words of White’s introduction to Chapter 5 perfectly capture the interplay between the mind and the page:

     "The mind travels faster than the pen; consequently, writing becomes a question of learning to make occasional wing shots, bringing down the bird of thought as it passes by. A writer is a gunner, sometimes waiting in the blind for something to come in, sometimes roaming the countryside hoping to scare something up. Like other gunners, the writer must cultivate patience, working many covers to bring down one partridge.” 

Any mathematician knows just how fleeting a moment of inspiration can be. ‘The Elements of Style’ trains the writer,  but more generally the thinker, how to refine his aim, to recognise a fleeting thought that ought to be timeless, and, by capturing it in the form that best reflects its content, to bring it to fulfilment.  

Modelling hierarchies

In a discipline otherwise built upon the tenets of objectivity, mathematical modellers are a bit of an anomaly. Most, including myself, will admit that mathematical modelling is as much of an art as it is a science. Art, however, still relies on a certain order. Whether or not they do so consciously, mathematical modellers often follow some hierarchy of reasoning to aid in the development of their models. I’d like to highlight two of those hierarchies here. 

The first is a hierarchy of data analysis by Jeffery Leek and Roger Peng, published in the 20th March 2015 edition of Science. It’s worth reading the full article (it’s brief), or at least glancing at the flow chart that offers a dichotomous key for analysing data. For those who cannot access the article, I’ll summarise the approach here (with a bit of interpretation, so that makes this an Exploratory Analysis!): 

  • A summary of data without interpretation is a descriptive analysis.
  • Add in some interpretation, and you get an exploratory analysis.
  • If the interpretation quantifies how the data generalises to new samples, it’s an inferential analysis.
  • Speculation on how the data might apply to new individuals makes it a predictive analysis. 
  • Quantifying how a change in one measurement affects another, without regard to precisely how the change takes place, gives a causal analysis.
  • And, quantifying the process by which a change in one measurement affects another constitutes a mechanistic analysis. 

These categories aren’t totally disjoint, nor do they necessarily flow linearly one from the other. While this list places analytical strategies loosely in order of level of inference, it’s not a ‘ladder' that should be climbed at every opportunity. The available data, and the central question to be answered, will indicate which type of analysis is most appropriate. 

The second hierarchy relates specifically to spatial data analysis. It nests within the mechanistic analysis step from above. These ideas are from Prof. Aaron King, drawn from a presentation he gave in September 2015 during a workshop on human mobility models. All (mis-)interpretations of his words are my own. Given data of the format

y1(t1) y2(t1) … yn(t1)

y1(t2) y2(t2) … yn(t2)

.

.

.

y1(tk) y2(tk) … yn(tk) 

at n spatial locations and k time points, how do we characterise it? He proposed building successive models according to the following criteria, and stopping once a step up in the hierarchy yields no better description of the data:

  • independent dynamics, distinct parameters (each location follows its own rules)
  • independent dynamics, shared parameters (locations share the same rules, but otherwise don’t communicate)
  • independent dynamics, shared parameters, spatial covariates (the rules a location follows depend on its location in space)
  • independent dynamics, random effects (same as above, but we account for unobserved factors by allowing some random variation in those rules)
  • full spatio-temporal model (locations directly influence one another)

The guiding principle behind this hierarchy is embeddedness in real space. At the first level, the locations might as well be completely disconnected. At the second, we recognise that they live in the same world, but claim that their spatial position doesn’t otherwise matter. At the third, we begin to account for spatial variations. At the fourth, we reluctantly admit that we don’t know everything about the space in which the locations live, so we allow the model to vary some, and therefore to clean up the messes where our powers of explanation have fallen short (did I mention that this is an art?). At the fifth, we assert not only that position matters, but also that dynamics in one location propagate to the next, again with some random variation.

Hierarchies like this one help to avoid the temptation to connect things that aren’t necessarily linked - a temptation that is especially strong in spatial analysis. It’s second-nature to assume that two events that take place near one another, in space and/or time, must somehow relate. However, if we can explain the phenomenon without recourse to such a link, then we ought to accept the simpler explanation. 

Work of the week: UK newspapers’ representations of the 2009-10 outbreak of swine flu: one health scare not over-hyped by the media?

There’s little doubt that emerging infectious diseases can elicit a major public response that is reflected in and/or driven by the media. This paper[1] questions the validity of an accusation often leveled at media outlets during outbreaks: that news coverage exaggerates public health risks. According to authors Hilton and Hunt, there was no such exaggeration during the 2009 H1N1 pandemic, at least for UK print sources. They analyzed thousands of news articles printed in 2009 and 2010, recording key ideas and general tone. They conclude that, on the whole, the media faithfully communicated up-to-date scientific understanding of the disease and of the public health risk it posed.

This article brings a number of valuable points to the discussion surrounding a quantitative history of the 2009 pandemic. First of all, the authors offer a sound framework for analyzing the qualitative data generated by media sources. This information, in turn, can help explain anomalies in data sets that aim to measure flu activity more directly, yet are still susceptible to public panic. Second, though not mentioned by the authors, their work encourages scientists to maintain communication with media sources - we need not despair, for it seems our findings are usually  faithfully communicated. It’s encouraging to see such a thoughtful analysis of a contentious topic, and even more encouraging that the results offer a strong counter to the cynicism that can dominate these discussions.

References:

[1] Hilton, S., & Hunt, K. (2011). UK newspapers’ representations of the 2009-10 outbreak of swine flu: one health scare not over-hyped by the media? Journal of Epidemiology and Community Health, 65(10), 941–6. doi:10.1136/jech.2010.119875

Work of the week: My students’ Mathematical Biology problem sets

I met this week with nine undergraduates, all of whom have elected to take the DAMTP’s course in Mathematical Biology. They came in pairs (except one), and with them a deluge of problem sets to mark. On the whole, their math was good, but one bit was largely missing: interpretation. The problems had to do with real-world systems, such as competing insect populations and over-fishing in lakes, but their answers looked more like “period = 4 +/- 2n” than “Stop fishing now!!!”. I assumed, initially, that it must have been a simple lack of time that prevented them from writing the implications of their answers on paper. These students are busy, after all. When I met my supervisees in person, however, I realized that interpretation had taken such a mental back seat that it somehow missed the bus entirely. 

One student made a comment that particularly struck me. A self-proclaimed "pure-mo" (pure mathematician), his mathematics was pristine, but I was forced to leave a glaring “Interpretation?” with my green pen beneath each of his answers. During his supervision, it took real effort to coax real-world ramifications from him ("how would this affect a strategy to re-stock a lake? to treat a blood disorder? When can we guarantee the rabbits will survive?” - silence), not because he was unwilling to work with me, but because he missed some crucial link between the figures on his page and the outside world. “I’m afraid the course might get too applied,” he confessed, clearly referring to his difficulty in interpreting answers. As someone who has spent most of his mathematical career with a fear of the opposite sort, that gave me pause. What could make this student so averse to interpretation?

I think the answer lies in a particular - and problematic - understanding of mathematics as a whole. To many of the students with whom I’ve worked, solving a problem is fundamentally a sort of ‘closing.’ Mathematics to them is a sport, a set of puzzles to be worked out. The fact that we might derive some significance from those results is a passing curiosity, accidental to the game itself. When the problem is solved and the answer boxed, they are done, and can move on to the next contest. I argue that the match hasn’t even begun - that the real mathematics happens when they sit back, take a moment to let the result sink in, and realize what their boxed answer means. The best mathematicians I’ve known, both pure and applied, are masters of interpretation. True, the pure mathematician prefers to answer ‘how does this reshape my understanding of mathematics?” while the applied mathematician tends to answer ‘how does this reshape my understanding of the world?” but the distinction, I argue, is of secondary importance, and indeed, all mathematicians should hold both questions in mind. Interpretation, of any sort, inverts the problem-solving act from a closure to an opening. By interpreting a result, we not only strengthen our foundation of understanding, but we also open ourselves up to the possibility of being surprised - surprised by, say, an inconsistent result, or an unexpected similarity to another field. This opens up new questions, new possibilities, new modes of thought, and ultimately breathes life into a field. 

The beauty of Mathematical Biology, and of applied courses in general, is that physical explanations are often the easiest ways to entice students to begin the practice of interpretation. They care about the world in which they live, so when they realize that an esoteric math problem can help them better understand their surroundings, they get hooked. I’ve seen the fire light in their eyes, and it won’t be going out soon. We who have taken a responsibility to educate students need to bear in mind that our role is not only to help them master theory, but to assist them in drawing ever-deeper meaning from the questions they answer, and in that way, to fan the flames.