by John Allen Paulos
Intelligibility or precision: to combine the two is impossible. ―Bertrand Russell.
Please forgive the long letter; I didn’t have time to write a short one. ―Blaise Pascal
I have always resonated with the two quotes above and believe they’re particularly germane to writing popular mathematics. Let me start with Russell. If his remark is taken literally, I would disagree with it, but if we take it merely as pointing out the often inevitable trade-off between precision and intelligibility, I find it rather profound. In my books I’ve certainly tried to be both precise and intelligible and hope that for the most part I have succeeded, as have so many other popular author of mathematics. The fact remains that combining the two is often a difficult task that at times depends on extra-mathematical understandings.
Take, for example, the notion of a continuous function in mathematics. Perhaps as a first approximation we might say that a function is continuous of we can graph it without lifting our pencil off the page. No breaks. This is intelligible, but is hardly precise and is, in fact, not what we mean by a continuous function. As generations of calculus students have understood (or misunderstood), the standard definition is simply not intuitive, involving as it does a complex statement involving the function in question and deltas and epsilons, Greek letters measuring distances along x- and y-axes. Unfortunately, immediately insisting on precision is a good way to discourage students from taking calculus.
By contrast, consider the Banach-Tarski theorem, which states that for a solid ball in three-dimensional space, there exists a way to decompose it into a finite number of non-overlapping subsets, which can then be rearranged and put back together so as to yield two identical copies of the original ball. The proof can be made precise and bullet-proof, but can one make it intelligible? In this case, only with great difficulty. Incidentally, the theorem is sometimes referred to as the Banach-Tarski paradox.
Precision sometimes comes for free. This is the case when a single insight can make what seems like a recalcitrant problem completely intelligible and without any particular need for precision. Elementary combinatorics is one source of such problems. For example, can you change some of the plus signs in +1+2+3+4+5+6+7+8+9+10 to minus signs to make the sum equal to zero? How, or why not? Answer in below.
Other issues arise when the mathematics requires some social understandings, as is the case with probability. The subject frequently gives rise to demonstrations that are almost unintelligible, at times necessarily imprecise, as well as counter-intuitive. Of course, what is intelligible and precise depends on the eye of the beholder. A classic extended example that has led to a whole literature concerns a woman who has two children at least one of whom is known to be a boy. Given this knowledge, what is the probability she has two boys? The answer is subtly dependent on whether the boy or the family is randomly selected.
If the family is randomly selected, this probability she has two boys is 1/3. Why? Note that there are four equally likely kinds of two-child families, those having two boys, an older boy and a younger boy, BoBy; those having an older boy and younger girl BoGy; those having an older girl and younger boy GoBy; and those having two girls GoGy. Since only one of the three types of family having at least one boy has two boys, the probability the woman has two boys if we know she has at least one boy is 1/3.
Yet, there is a rationale for the common belief that the probability the woman has two boys if it’s known she has at least one boy is “really” 1/2, not 1/3. How is this so? If we randomly choose a boy from families with two children at least one of whom is a boy, then there are four possibilities for the type of boy we might choose. It might be the older boy in the family with two boys BoBy, or the younger boy in the family with two boys BoBy, or the boy in the older boy/younger girl family BoGy, or the boy in the older girl/younger boy family GoBy.. Two of these four possible types of boys come from families with two boys at least one of whom is a boy. Thus, the probability the woman has two boys given she has at least one boy is 2/4 or 1/2.
So which is it? 1/3 or 1/2? It hinges on a number of factors, two of which are as described. How it becomes known that there is at least one boy in the family is another complicating factor. There have been, as noted, many articles and controversies that have arisen from this two-boys problem (or, to give the fictional mother and her offspring a break, from the identical two-girls problem and involving the father).
Two more related, but quite different problems, one easy, one quite counter-intuitive. The easy one: If a woman has two children the older one of whom is a boy, the probability she has two boys is 1/2. The reason is that there are only two equally likely types of two-child families, BoBy or BoGy.
And the counterintuitive one: If a woman has two children at least one of whom is a boy born in summer, the probability she has two boys is, rather surprisingly, 7/15. Why? The precise number can’t be easily explained without constructing the relevant sample space (list of possibilities), but we need no elaborate theory involving global warming or genetics or anything else to account for the above probability. It’s simply a consequence of quite plausible assumptions.
Applications of mathematics, especially in probability but also in statistics, often have significant social consequences and are vulnerable to significant social misunderstandings. Depending on one’s audience and intent in writing, the weight we give to intelligibility and precision will vary. Since a number of my books are intended for an intelligent, but not mathematically sophisticated audience, I have generally opted for intelligibility over precision, although at times, as in the two-boys example above, these characteristics are not incompatible, but almost indistinguishable.
I also mentioned Pascal’s quote on brevity above, which has been variously attributed to a number of authors, but like Russell’s remark it is, I think, rather profound. Relevant perhaps is the fact that mathematicians aren’t generally impressed with a new 100-page proof of a theorem when there’s a trenchant insight that leads to an older and succinct 3-page proof. Moreover, the shorter proof is likely to be more intelligible and more precise.
Brevity is undervalued and being brief is a skill that more people, not only mathematicians, should cultivate. To that end, I used to give the following assignment to students in an elementary course I taught. I would verbally describe at some length some situation, some mathematical parlor trick, or some puzzle, and then ask the students to write up an explanation that mentioned and explained all the salient points I made in my description. If their written explanations did that, they would receive at least a C on the assignment. To get an A or a B, they had preserve the essence of their explanations, but use fewer words, the fewer the better. This I found made them think more clearly about what was essential and primary and what were the logical connections among the various points.
After a while the students improved at this task as their expositions became tauter and more logical. I admit that being brief is perhaps an occupational hazard of being a mathematician and author. I don’t think being brief would help a writer of romances or soap operas very much. In any case, I have always disliked expository writing that began, for example, with a character gazing out over a lake and remembering some long-ago events, at best marginally relevant to the exposition. Similarly, I’m not a fan of an exposition beginning with a couple of character sketches, also tenuously connected to the topics of the exposition.
Instead there’s often bias in the direction opposed to my predilections. Details are consigned to an appendix and the body of the text consists of vacuous tales faintly associated with the topic being explicated. I’m mocking and exaggerating, of course, but I do frequently think to myself, “Where’s the beef?”
In my opinion, intelligibility, precision, and brevity are three (among many, including liveliness) desiderata of good writing on popular mathematics. Once the logical skeleton of an account has been written, then representative examples, stories, anecdotes, vignettes, and jokes (or maybe especially jokes) can be added at the beginning of the discourse or wherever it’s appropriate to do so. But they too, especially if they’re illustrative of the topic of the exposition (as jokes often are), should be written intelligibly, precisely, and briefly, even though the intensional logic of stories and narratives is quite different than the extensional logic of statistics and numbers. But that’s a different story.
My apologies for aspects of this piece that are incomprehensible, vague, and long-winded, but neither Russell nor Pascal is around to speak for themselves.
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Answer: The sum of the first 10 integers is 55, an odd number. And any plus sign changed to a minus sign decreases the sum by an even number (why?), which results in another odd number. (Odd minus Even is Odd.) So decreasing an odd sum by a sequence of even numbers will never result in the even number 0, but will always result in an odd number.
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John Allen Paulos is a Professor of Mathematics at Temple University and the author of A Mathematician Reads the Newspaper, Innumeracy, and a new book, Who’s Counting –Uniting Numbers and Narratives with Stories from Pop Culture, Puzzles, Politics, and More. (the link to the Amazon page of the book.)