[A series of posts explaining a paper on the mathematical modeling of the spread of antiviral resistance. Links to other posts in the series by clicking tags, "Math model series" or "Antiviral model series" under Categories, left sidebar. Preliminary post here. Table of contents at end of this post.]
The Modeling Series (click Math Modeling Series under Categories in the left sidebar) is a moving target for me. Even though the first draft of the post you read each day was written at least three weeks earlier, each has also been freshly worked on as I go back and tinker and adjust and try to improve my explanations or find holes in the exposition. By this time I already know my audience is shrinking as we go deeper into the details of a specialized scientific paper. But it has become a personal challenge to see if I can explain even the details to a readership I know is interested in the subject, motivated and intelligent (as well as a little crazy). If I can’t do that, I’m not much of a teacher. So I feel compelled to pause for a moment to discuss the challenges for non-specialists in understanding a mathematical modeling paper. This pertains not just to non-scientists, but to many talented and brilliant scientists who shy away from the mathematics and resignedly “take it on faith.”
“Mathematics” isn’t hard. We all use it constantly when we file our taxes or count our change at the store or read public opinion polls. But some mathematics is hard. Very hard. This isn’t surprising. Making “music” isn’t hard, either. Many of us do it all the time, even if it’s only to enjoy it or sing along in the car. But making music at the level of a professional musician is hard. Very hard. I’ve played a number of musical instruments, strictly amateur and very badly. A couple of them were fretted string instruments (5-string banjo, guitar). I love violin music and in my middle age decided I’d take it up. How hard could it be? After all, I already know how to play some string instruments. For an adult beginner who never touched a fiddle in his life, the answer to that question turned out to be that it was unbelievably hard. It felt like an exercise in “how many things can I think about doing all at once.” Although I eventually was able to play full classical repertoire in the second violin section of our community symphony orchestra, I was still so bad I couldn’t stand to hear myself play and finally gave it up. Meanwhile, ten year olds were playing violin after two years ten times better than I did after five. Of course their mothers made them practice. That helps.
What’s this got to do with reading a mathematical modeling paper? In the next post of the series (part IV) we will start the process of “thinking mathematically.” There will be no formulas or equations. Still, it will be very hard for some of you, and you will be tempted to give up. I hope you won’t, so I want to try to guess at the source of some of your difficulties. It isn’t because you are mathematically inept or not “mathematically talented” or smart. On the contrary, I would suspect for some of you it is because you are very smart and insist on asking certain questions that get in your way. Like, “Why?” You want to know how it “really works” or you see all sorts of problems with what’s been said. You will read the mathematical reasoning and feel you “don’t get it.” You must be missing something. You probably are, but it isn’t what you think.
The best explanation I know has been attributed to one of the greatest mathematical physicists of the 20th century, John von Neumann:
“In mathematics you don’t understand things. You just get used to them.” (von Neumann, attributed)
Like a ten year old playing a violin. After a while, she stops thinking about why she used fourth position instead of third to play that note. She just did it. She got used to the fact that fourth position generally works out better in that passage because of the notes that came before and afterward, or because she or her teacher preferred it, or her fingers were long or short or fat. Those are “reasons” but they aren’t forced by a mysterious logic you don’t understand. A lot of the applied mathematics is like that (in pure mathematics you are forced by logic, but there is no claim it relates to the real world question of “why?”). After a while, you get so used to it you think you understand it. But if von Neumann can admit he never really understood it but just got used to it, then the rest of us can also feel somewhat relieved that maybe our discomfort at not understanding Hilbert space the way he did is because we haven’t spent as much time living there as von Neumann did.
Don’t get me wrong. I’m not saying we are all potential von Neumann’s any more than the average ten year old is a potential Heifitz on the violin. But almost all of us are capable of appreciating Heifitz or Doc Watson or Earl Scruggs or any other supremely gifted musician we couldn’t hope to become. We can even make a claim to “understanding” them. In the same way, you can appreciate what mathematical modelers are doing in the Lipsitch et al. paper we are discussing. What will help is to suspend your discomfort at not “understanding” everything long enough to start the process of getting used to it.
On the other hand, there are reasons applied mathematicians do things. For example, I will try to explain the reasons that led them to guess that the Law of Mass Action might be a useful tool in epidemiology (part IV). It was speculation at first but experience has shown that despite many reasons you might think it wouldn’t work, for many purposes it works extremely well. Once they saw that, they got used to it. Fast.
The bottom line here is that I hope you give yourself a chance to get used to it, too. Keep reading. Modeling research has been maturing for a century with fits and starts and blind pathways, as in any science, but it has made great progress, with the help of wonderful new mathematical research and powerful new computational tools. Molecular biology is a younger science and in ten years will probably look very different than it does now. Yet we wouldn’t ignore today’s advances because we know tomorrow we will be smarter.
Enough pep talk. In part IV we start uncovering the mathematical machinery in Lipsitch et al.
Table of contents for posts in the series: