After the show aired, I received a few letters of complaint from statisticians, who lamented my use of the phrase "Bayesian mathematics" instead of the correct term "Bayesian statistics".
In fact, my choice of words was made with some care. Although the Math Guy segments are genuine conversations, they are not entirely random. True, there is no script, other than the introduction that the show's host, Scott Simon, reads out, and a list of possible question prepared in advance - a list he often abandons as soon as the conversation has begun. Nor do we rehearse beforehand. But without any prior planning, there would be little chance we'd be able to get some fairly sophisticated mathematical ideas across in the 7 to 10 minutes we generally have available.
The Math Guy pieces work in large part, I think, because of what happens before we ever walk into the studios. In particular, I think about how best to get the main idea across. (In a short radio segment in a magazine show aimed at a general audience, the most you can aim for is one main point, and it has to be made using everyday language the average listener is familiar with.) Since I do not know what questions Scott will ask (except in general terms), this amounts to being clear in my mind what that one main point is, asking myself what kinds of example I might have to bring in to make my point, and thinking about words to use (or words not to use) to ensure that the message really does reach the average listener.
I'll be brutally honest. I don't worry about how the experts will react. I figure that they are a vanishingly small proportion of our listeners, and besides, they don't need me to inform them what's going on in their field anyway. Here is why I made a conscious decision in advance to use the phrase "Bayesian mathematics" rather than "Bayesian statistics" in my piece on May 18.
The main message I wanted to convey was that there is a powerful method, based on a mathematical theorem proved over two hundred years ago, for combining human judgements with often massive amounts of statistical data in order to produce improved versions of those initial judgements. To the experts, that method is "Bayesian statistics." I reasoned, however, that to most people "statistics" is not a process of getting information from numerical data (which is what professional statisticians, and I for that matter, mean by the word). Rather "statistics" is generally understood to mean tabulated numbers, pecentages, and the like. Thus to the layperson, the phrase "Bayesian statistics" will be understood as a certain collection of numbers, possibly the scoring averages of a little known baseball team called "The Bayes". On the other hand, everyone knows that "mathematics" is a process for handling numbers. (This is also false, but, as Walter Cronkite used to say, that's the way it is.) Thus, I felt, using the term "Bayesian mathematics" would convey to the audience a much better general sense of what Bayesian statisticians (sic) actually do than would using the term "Bayesian statistics."
Now, let me repeat, we're only talking a 7 or 8 minute radio slot here, so much of the above thinking was done as I cycled across the Stanford campus to the studio. (The conversation generally takes place with Scott in the NPR studios in Washington, D.C. and me in the campus radio studio at Stanford.) I did not go out and systematically check if my reasoning was correct. I just relied on my instincts and what experience I have gleaned both in the classroom and through my efforts to raise the public awareness of mathematics. Still, a quick sample poll among two or three nonmathematical friends I carried out afterwards (yes, I know this is not statistically valid!) confirmed what I had suspected in this case. The term "Bayesian statistics" conjurs up an image of a particular set of numbers, whereas "Bayesian mathematics" suggests some sort of computational process.
So what exactly is my point? Just this. As a mathematician, I agree that correct terminology is important in mathematics and statistics - for those who practice those disciplines. In the classroom, I regularly stress the need for precision. (Although the degree to which I do this depends on the kind of course I am teaching. I am far more demanding with a class of mathematics majors than when teaching math to business students.) But what about everybody else? - by far the greater majority of the population.
The aim of any kind of teaching, surely, is to help people advance their understanding. That can only be achieved by starting with concepts and ideas the student already has and building upon them. In mathematics, more so than in any other discipline, the words we use are particularly critical, since mathematical concepts are entirely abstract, and can only be approached through the words we use to describe them. The student cannot, in general, rely on knowledge or intuition of things in the familiar, everyday world. When we mathematicians use words such as function, relation, continuous, derivative, ring, field, etc. (and, let's not forget, "mathematics" and "statistics"), we mean something very different from what those words conjure up in the general populace.
Now, in the classroom, we have a chance, over a semester, as we check the assignments handed in to us each week, to help the students develop an understanding of what we mean by those terms. But in a one-off lecture to a general audience, or in a radio broadcast, that is not possible. We cannot get away from the fact that any word we use will conjure up, automatically, an idea in the listener's mind. And we have virtually no chance of getting beyond that.
Giving a brief explanation of what we mean is unlikely to be successful. After all, appreciating the distinction between the mathematical meaning of the word "relation" and the everyday meaning requires an understanding of both meanings! Moreover, an aside to explain terminology diverts attention away from the main message we want to convey.
Instead, we have to find a way to tell our story in terms of the meanings our audience automatically attach to the words we use, generally the everyday meanings of those words.
In my Math Guy contributions, I have played fast and loose with terminology from physics, computer science, aerospace engineering, psychology, and probably some others areas as well, and now also with statistics. Yet the only groups who regularly complain - and it is regular - are the "math types," those for whom correct terminology is crucially important to practitioners, and who see it as important that their students use the right words. I believe that this excessive focus on using the correct terminology, while important in the classroom (at least, some classrooms), is one of the reasons why we mathematicians have not been as successful as other sciences in explaining to the general public exactly what it is we do and why it is important.
When it comes to getting "big picture" messages across to the general population, I think it is a mistake to focus on the details. The layperson doesn't give a hoot what the terminology is. But (to take the example that I started with) learning that there is a way of combining expert guesswork with masses of data to make predictions about possible terrorist attacks? Boy, that's cool stuff. (I know that, because I spent the first three days of the week following my Homeland Security piece fielding phone calls from journalists who had heard the show and wanted to know what "Bayesian mathematics/statistics" was and how it played a role in Homeland Security.) Who, apart from some professional statisticians, cares about what words those professionals use when talking among themselves?
And, of course, the same point applies, to differing degrees, whenever we teach mathematics to different kinds of students.