Mathophobia

In many fields of science and engineering, any technical argument must be formulated mathematically if it is to be taken seriously. In contrast, in popular writing--even about science and engineering--including a single equation is a known recipe for getting many readers to click on to the next story. Understanding this discordant reaction to math reveals a lot about how popular and technical writing differ.

Back in 2003, when I was thinking about morphing from a practicing scientist into a science writer, I seriously questioned how I could possible to explain scientific arguments without variables and equations. Without the precision of a mathematical description, I wondered, how could I really know whether readers interpreted ordinary English phrases the way I intended? Moreover, without a algebraic description, how could readers judge how well a model matches observations?

Interestingly, in the years since, I've almost never felt hobbled by not being able to explain things with equations.

A lot of the difference arises from the different goals of journal articles and popular stories, and their very different sources of authority.

In a journal article, the goal is to convince other experts. In other words, the article should ideally be self contained, assembling all the relevant details so that an independent observer can make up their own mind.

In contrast, a popular science story aims merely to describe the conclusions, not prove them. As David Ehrenstein, the editor at Physical Review Focus, once told me, the goal is to present a plausibility argument for the conclusions: to give enough context and explanation that readers can appreciate what's being claimed and who's claiming it.

The last point is also critical: by and large, popular writing gains its authority from the quoted judgments of experts, not directly from the model or observations. Indirectly, of course, this authority comes from the reputation of the writer and the publication (that is, the editors), because they are the ones who decide which commentators are worth quoting. (Of course, those commentators must also respond to emails or phone calls!)

Since the goal is plausibility and a qualitative understanding, the limited precision of ordinary writing is usually good enough to convey the message.

The Wisdom of Ignorance

Physics and math demand a certain mode of thought. Lots of people think that doing well in those classes takes intelligence, and that's part of it. But they also require something else that is not always a good thing: comfort with abstraction, or stripping problems down to an idealized cartoon.

Those of us who excelled in these subjects can be a bit smug towards those who didn't, but replacing real life with a cartoon isn't always a good thing. In addition to hindering social relations, it can obscure important truths.

It's interesting to contrast Aristotle's view, for example--that objects in motion naturally come to rest--with Newton's--that they naturally keep moving. Thinking about familiar objects, you have to grant that Aristotle had a good point. Of course, he'll leave you flat: if you figure out how to include friction, Newton is going to get you a lot further--even to the moon. But beginning students are asked to commit to an abstract formalism that has a stylized and flawed relationship to the world they know.

Probability has a similar problem.

Most normal people, for example, expect that after a flipped coin shows a string of heads, tails is "due." Probability theory says otherwise: the coin doesn't "know" what happened before, so the chances on the next flip are still 50/50. Abstraction wins, intuition loses.

[Actually, Stanford researchers showed in 2007 (pdf here) showing that unless a coin is flipped perfectly, the results will not be 50/50: if the coin is spinning in its plane at all, its angular momentum will tend to keep it pointed the way it started out. But that's a minor issue.]

On the other hand, there are lots of cases where common intuition is "directionally correct." Take another staple of introductory probability courses: a bag full of different-colored balls. In this case, the probability won't stay the same unless you put each ball back in the bag after you choose it. If you keep it, choosing a ball of one color will increase the chances of a different color on the next pick, in keeping with intuition.

Of course, intuition doesn't get the answer with any precision, and it gets it completely wrong for the coin flip. To do it right, you need the abstract formalism. Still, it's easy to imagine that our brains are hard-wired with an estimating procedure that gets many real-world cases about right.

In other cases, our intuition is more flexible than slavish devotion to calculation. Suppose I start flipping a coin. It's not surprising to see heads the first time, and the second time. How about the third time? The tenth? If it keeps coming up heads, you will quickly suspect that there's a problem with you original assumption that the probability is 50%. Your natural thought processes will make this shift naturally, even if you might be hard pressed to calculate why. Probability theory is not going to help much when the assumptions are wrong.

It's true that people are notoriously bad at probability. ScienceBlogger Jason Rosenhouse has just devoted an entire book to one example, the "The Monty Hall Problem: The Remarkable Story of Math's Most Contentious Brain Teaser. (It was also discussed in 2008's The Drunkard's Walk, by Leonard Mlodinow, and in The Power of Logical Thinking, by Marilyn Vos Savant (1997), the Parade columnist who popularized it.)

The Daily Show's John Olliver amusingly explored how simple probability estimates (especially at around minute 3:20) help us misunderstand the chances that the Large Hadron Collider will destroy the world.

Still, our innate estimation skills developed to deal tolerably well with a wide variety of situations in which we had only a vague notion of the underlying principles. Highly contrived, predictable situations like the coin flip would have been the exception. Even though our intuition frequently fails in detail, it helped us survive a complex, murky world.

Spoiler Alert

On this election day, many New Jersey voters faced a common, unsatisfying, and ultimately unnecessary choice: vote for the best candidate, or vote for a candidate that the polls suggest could win?

Why unnecessary? In a reasonable electoral system, voting for the person you prefer would not increase the chances for someone you don't. But that's not how we do it.

A typical vote today awards the election to the candidate with the most votes--a plurality, not necessarily a majority. If a third candidate enters a two-person race, he or she is more likely to draw votes from the more similar candidate. The result is that an extra candidate with a particular viewpoint can reduce the chances of that viewpoint prevailing. This is the "spoiler effect."

The best known example is Ralph Nader's entry in the 2000 election in Florida. Although Nader got only a few percent of the votes, if everyone who voted for him had voted for Gore, Gore would have beaten Bush--in Florida, and in the country.

But there's nothing liberal or conservative about the problem. Any candidate can lose because of a strong third-party candidate with similar views.

This is just wrong.

It should be changed.

And there is a simple way to change it.

In instant runoff voting, voters rank all the candidates, rather than just voting for their first choice. Not too hard, right?

If no one gets a majority of first-choice votes, the candidate with the fewest is eliminated. Those ballots are redistributed to the second choice on each ballot. No votes are "wasted," but voters get to express their true preference.

The results are exactly to what would happen in an ideal runoff election, except that there's no need for the expense and low turnout of a second election.

In 2000 Florida, for example, Nader would have been eliminated, and the ballots that listed him as first choice would have been allocated between Bush and Gore, depending on who people listed as their second choice. In 2009 New Jersey, independent Chris Daggett was expected to draw more votes from the Republican candidate, which could tip the balance to the Democrat.

Many individual cities, like Oakland and Memphis as well as some national elections use instant runoff voting now. There's nothing particularly difficult about it, except that the two dominant parties may see it as a threat to their exclusive right to power.

Instant runoff voting isn't perfect. Pretty much all voting systems have some quirks that sometimes give results that seem obviously wrong.

But it's not nearly as bad as what we have now.