GEN-SPEC.

THE MATHSEMANTIC MONITOR [*]

ONCE UPON A TIME there was a country with two languages, a general language, Gen, and a specialized language, Spec (rhymes with peck). Almost all adults in the country used both Gen and Spec to some degree. Most people used Gen reasonably well and some had become really good at it. Not as many people used Spec reasonably well and only a small number had become really good at it. Some people could use both Gen and Spec reasonably well so long as the two languages were kept separate. But few people were any good at combining Gen and Spec.

People who knew Gen reasonably well communicated easily with their neighbors, because they used the same Gen. They could not, of course, communicate easily with people in other countries if they happened to use a different Gen. This was the usual case, as in the world at that time there were thousands of different general languages, Gen1, Gen2, Gen3 ...

It was different with the people who knew Spec. They could communicate in Spec with people from any country, because Spec was still Spec in every country. Spec users could not, however, and unfortunately, say much of anything in Spec about anything that mattered for life in general, because Spec was a truly specialized language. Therefore, except for the times when highly trained specialists discussed matters dealing entirely with Spec itself, few people ever communicated entirely in Spec.

Almost every school in the two-language country taught Gen and had done so from the day it first opened its doors. The situation with Spec was different. At first only a few colleges taught Spec. But as Spec's importance grew, the country's educators and educational administrators couldn't help but notice. With their help, colleges began to judge applications for admittance in part on a student's competence in Spec. This led after two hundred years to Spec's being taught in all grades from kindergarten up.

Among the few aspects of life that Spec communicated really well were rates of change and how different kinds of change related to each other. As the country grew and changed, and as its leaders and people began to see that the changes interacted with each other and were at least partly subject to human control, Spec took on ever more importance.

Spec alone, however, wasn't much help. One still had to use Gen to identify different kinds of change (such as those involving personal and social habits, science and technology, economic and political forces, and all their institutions) and to discuss whether they were desirable changes or not. One needed to be really good at both Gen and Spec to analyze how these things interacted and to determine what, if anything, one should do about them. One also needed to be reasonably good at both Gen and Spec even to understand the analyses.

Unfortunately, as already said, few people were any good at combining Gen and Spec. In spite of the apparent need for some true Gen-Spec experts and reasonable competence in the population at large, there were no university courses in Gen-Spec and no professors of Gen-Spec. There were also, as a result, no primary or secondary school teachers or students of Gen-Spec. Indeed, the very idea of combining Gen with Spec education was almost unthinkable. Gen and Spec had evolved separately as academic disciplines. Each had its own faculty, its own budget, its own heroes, its own journals, and its own clubs. They were academically separate and distinct. They didn't mix.

The language separation found in the two-language country was actually a lot like that described in The Phantom Tollbooth [1], even though this story by Norton Juster is a children's fantasy peopled with make-believe characters. For example, chapter three brings the young hero, Milo, and his friends to a place the gateman announces as "Dictionopolis, a happy kingdom, advantageously located in the Foothills of Confusion and caressed by gentle breezes from the Sea of Knowledge." Later, at a banquet, King Azaz the Unabridged asks Milo whether he can sing songs, tell stories, or compose sonnets. On being told not, Azaz asks, "Can't you do anything at all?"

"I can count to a thousand," offered Milo.

"A-A-R-G-H, numbers! Never mention numbers here. Only use them when we absolutely have to," growled Azaz disgustedly.

Milo soon discovers that his mission is to obtain the return of the two Princesses, Rhyme and Reason, and that this will require dealing with the Mathemagician. The Humbug (a bug who hums) then spells out the difficulties. Even to reach the Mathemagician, he says, would require Milo to "travel through miles of harrowing and hazardous countryside, into unknown valleys and uncharted forests, past yawning chasms and trackless wastes, until he reached Digitopolis (if, of course, he ever reached there)." One can get a flash of the separation of Gen from Spec in the distance of Dictionopolis from Digitopolis, so vividly portrayed as MilO sets out in chapter 9.

Soon all traces of Dictionopolis had vanished in the distance and all those strange and unknown lands that lay between the kingdom of words and the kingdom of numbers stretched before them.

After five chapters of adventures in the between-lands, chapter 14 brings Milo the first sign (literally) of the number kingdom. It reads:

DIGITOPOLIS

5 Miles

1,600 Rods

8,800 Yards

26,400 Feet

316,800 Inches

633,600 Half Inches

AND THEN SOME

Milo soon finds himself bedeviled with questions from Digitopolis's Dodecahedron gatekeeper, and finally he balks.

"I'm not very good at problems," admitted Milo.

"What a shame," sighed the Dodecahedron. "They're so very useful. Why, did you know that if a beaver two feet long with a tail a foot and a half long can build a dam twelve feet high and six feet wide in two days, all you would need to build Boulder Dam is a beaver sixty-eight feet long with a fifty-one-foot tail?"

"That's absurd," objected Milo, whose head was spinning from all the numbers and questions.

"That may be true," he acknowledged, "but it's completely accurate, and as long as the answer is right, who cares if the question is wrong? If you want sense, you'll have to make it yourself."

Laying The Phantom Tollbooth alongside the two-language country, of course, immediately gives the Mathsemantic Monitor's game away. Gen and Spec, in the United States, at least, go by the names of English and mathematics. Everything reported for Gen and Spec -- their educational history, the scarcity of users able to combine them, their application to the problems of change, their separation, and the lack of their academic combination, a mathsemantics -- applies to English and math.

Mathsemantics lies neither in the realm of Dictionopolis nor of Digitopolis. It is neither Gen nor Spec, but the combination of both. It lies at the overlap of general language and math, of words and numbers, and draws on both.

Academically, mathsemantics corresponds to neither English nor math. Hence neither English nor math teachers teach it. Indeed, it appears that neither discipline knows how to teach it, which gives sufficient reason for neither to acknowledge it. What would people think, after all, if they found out that something civilized life now needs falls through a wide crack in the academic floorboards?

Anybody who's attended school in the United States knows that English and math are treated as completely separate disciplines. The separation is so obvious and so taken for granted that nobody questions it. Therefore, the fact that the separation has profound effects also usually escapes notice.

But except for the long separation of English and math, how could a respected scientific magazine allow one of its writers to start an article with three examples of bad mathsemantics? (italics added)

Since the beginning of the AIDS epidemic, researchers have consistently noted a strong connection between HIV infection and other sexually transmitted diseases (STDs). Those infected with an STD are at least two to five times more likely to acquire HIV if exposed to the virus through sexual contact, and an individual infected with HIV and another STD is more likely to transmit the HIV virus to an uninfected person. [2]

Take the first mathsemantic blunder, at least two to five. Which number should we believe? If we believe "at least two," then the "five" becomes superfluous. If we believe "at least five," then the "two" becomes impossible. Trying to save the range by treating it as a group (the two-to-five group) makes no sense. So we can't tell how to read the phrase. Why isn't this as obvious as that "ain't" ain't good English? The author and editor probably had lots of input on English usage (split infinitives, dangling participles, that sort of thing) and lots of math input, but never received any warnings to look out for mathsemantic problems.

The second mathsemantic blunder, times more, lies in its ambiguity. Some people read "two times more likely" as meaning "twice as likely," whereas others read it as meaning "three times as likely." Why isn't this ambiguity an English no-no? Why isn't it clear that the exactly quantifying and relational terms, at least, two, five, times, and more, lead us to expect quantitative and relational precision, but as strung together the words thwart the expectation? No handbook warns against this.

The third mathsemantic blunder, the words "more likely," actually appears twice and each time introduces an incomplete comparison. It begs a "than." To understand the comparisons we must know more likely than what, when or who. Do we read the first comparison as, "Those infected with an STD are ... more likely to acquire HIV if exposed to the virus through sexual contact" than when exposed through non-sexual contact? Or do we read it, "Those infected with an STD are ... more likely to acquire HIV if exposed to the virus through sexual contact" than those not infected with an STD? If the latter (which seems more likely), then we can know to ask whether anyone has controlled the data for the fact that people with a sexually transmitted disease presumably have had more promiscuous sexual contacts. Similar considerations affects the second comparison with its "more likely" that leaves us dangling.

The Mathsemantic Monitor has scores of similar examples from respected scientific sources. He now pays more attention to them than to examples in the general press, because he's learned to take in stride the unending stream of mathsemantic mistakes in the general press. For example, he laughs off such howlers as:

Keep these tips in mind while training Spot or Fido: 1. Keep the lessons short. Four half-hour sessions will be more productive than one full hour. [3]

Somehow he expects more from the editors of Scientific American, Science, and Nature, to take just three respected journals that must combine English and math in every issue. He'd like to count them among those desiring to reduce innumeracy. Why, then, he asks, do they allow language like the following (italics added)?

But the only company ever to produce commercial magnetic RAM chips was Honeywell, and in 1997 its best devices were still 10 times slower, 256 times less dense, and far more expensive than DRAMs [dynamic-random-access-memory chips]. [4]

How can anything run 10 times slower? If it ran one time slower, it would stand still, wouldn't it? So, if it ran two times (let alone 10 times) slower, it would find itself beating a hasty retreat. Shouldn't the Scientific American insist that its writers say "one tenth as fast" and "one 256th as dense"?

The Mathsemantic Monitor recently visited his eye doctor, a man who passed from physicist to physician and who regularly asks after the Monitor's latest interests. On hearing the Monitor's complaint about "times more" ambiguity, the doctor paused and then said, "It's the same in Hungarian; it's illogical, but so what?"

"Often nothing," replied the Monitor, almost in tears as the doctor scanned his right retina with superbright light, "but consider the problem people have with percentages, where they don't know that a growth to 210% of a base figure amounts to a growth of 110% from the base. Do you see the connection?"

"Ah," the doctor said, switching his scan to the Monitor's left eye, the one with the incompletely detached vitreous humor, "you have a point. Usage hides the distinction."

"And I fear," said the Monitor while the doctor started to record his ophthalmological findings, "that the distinction will remain hidden until we focus on both math and English, or Hungarian, as languages and ask how we can put them together." The good doctor looked up quizzically but only smiled. The idea of treating math and ordinary language as two languages that we need to combine runs too much against the grain to permit quick understanding.

Take plurals. Plurals certainly form part of the explicit subject matter of English and the implicit subject matter of math. But into which domain do such questions as the following fall? "How many students are you if you study several subjects?" "If one person can take many trips, may we count passengers as people?" "Does it make sense to count book readings as books read?" "Does three times more than mean four times as many as?" And, of course, "can a device run 10 times slower than another device?"

"These aren't math problems," said the brightest senior in a statistics class after a mathsemantics presentation. "They're English problems." [5]

English teachers, apparently, wouldn't agree. Those with whom the Mathsemantic Monitor has discussed the matter see problems with numbers in texts as mathematical problems, not English problems. The head of one high-school English department even suggested that mathsemantic errors arise from unscrupulous efforts of advertisers and others to misuse numbers. She saw a problem in ethics, not in English.

The Mathsemantic Monitor once tried to interest a college in a combined math-English (mathsemantics) program. The math department took an interest but not the English department. The head of the math department felt such a program might lead students of English to take more of an interest in mathematical reasoning, even though he saw no need to address such problems in math class. The head of the English department saw no need to look into them at all.

So we have this strange circumstance: a clear set of language problems that involve both math and English for which neither discipline wants to take responsibility. Because the problems partly involve the other discipline, both disciplines disown them. They let these language problems fall between.

The Mathsemantic Monitor recently had occasion to write a mathematical educator about his review of Mathsemantics. The point was to reassure the educator, perhaps unnecessarily, that the Monitor did NOT hold math educators responsible for the poor state of mathsemantic understanding. The communication in its relevant parts went as follows. [6]

Standing as I have at the end of the assembly line, trying to make use of the product that has arrived at my office looking for employment, has given me a different point of view, I presume, from mathematicians or mathematics educators. I had hoped that college-graduate job applicants who thought of themselves as "good at numbers" would be ready to use numbers sensibly. Most of them who came to my door in Wayne, Pennsylvania, at least, were not ready. You might say that most of them had never thought about the kinds of problems I asked them to solve....

None of us, I presume, is happy that the applicants weren't ready. And here comes a critical point. The credit or blame for what I found usually gets placed on math education. My belief, however, is that English- (and liberal arts-) education, and the strict separation between them and math, is just as much, if not more, to blame.

Math educators could, it seems reasonable to believe, by incorporating some semantics into math, help pave the way for using math together with ordinary English when the student went into the world. But I wouldn't expect the instruction to make much headway without support from the English side. If English teachers don't realize that a "passenger" in the singular (to use just my well-worn example) is a person while "passengers" in the plural are not people but (usually) "trips," then English doesn't prepare pupils to deal with the plurals of the large and politically important class of nouns (customers, patients, voters, etc.) that treat an activity as a person. So I don't see this as just a math-education responsibility.

English deals with most things as individual instances. Journalists and writers in general are taught the power of the specific example. Some do very well with it, with what one might call the unitary view. Math, of course, deals far better than English with the non-unitary view, not only with plurals, but also with fractions, decimals, and higher-order functions. But what chance does math usage have of properly entering the mainstream of English-combined-with-numbers when students don't even regard math as a language?

English teachers, in the U.S. at least, see no difference (usually) between "three times as many as" and "three times more than," whereas logic dictates that the latter must be four times as many as. Doesn't this have some bearing on the difficulty of telling whether "150% more" in the press truly means that or only 50% more? How about "four times smaller"? (Astronomers here glibly talk about things being "thousands of times closer.") Doesn't that have some bearing on students unable to realize that a discount of 120% is absurd? Isn't math teaching to some extent being continuously undermined by what English and the liberal arts in general teach by their example?

Anyway, the point I want to stress is that I do NOT hold math educators to blame for the mathsemantic state of affairs in this country nor do I dismiss their efforts. I believe they often succeed in teaching what they set out to teach. I respect math educators for that. I'm glad they're working on improving math instruction.

Unfortunately, mending what I see wrong can't be done with math alone. It will take both math and English. That crosses guild boundaries, hence it becomes a truly difficult proposition.

Breaching departmental walls has a greater history of effort perhaps than of success. The Mathsemantic Monitor attended the University of Chicago when the idea of general education reached perhaps its highest vogue, so he tends to clip stories about its changing fortunes. Thus his files contain a clipping from the newsletter Observer, published by the American Psychological Society, reporting in its lead story that, [7]

"More than at any other time in the recent past, there is a demand for mechanisms and incentives to foster interdisciplinary research, education, and problem solving," wrote AAAS [American Association for the Advancement of Science] representatives in the Dec. 19, 1997, issue of Science. "Boundaries between scientific disciplines are collapsing, and the rise of interdisciplinary sciences is challenging the very concept of science as usual."

However, despite the enthusiasm for the cross-cutting research, there are a number of barriers that exist to hinder valuable interdisciplinary research from occurring.

The Mathsemantic Monitor trusts the last sentence comes as no surprise to anyone. Cooperation across departmental lines comes slowly. Nevertheless, an issue of Science, only a year and a half after the one quoted above, reports Lucy Shapiro, "a developmental biologist at Stanford University," as saying, "The convergence of chemistry, physics, biology, and engineering is upon us." [8]

Note that this convergence doesn't involve English or math. Indeed, it appears that math departments have a reputation for excessive inwardness.

A new report offers stern advice for academic mathematicians: Pay more attention to the world around you, or at least to the other departments in your university.

To learn how math departments view themselves -- and how they're viewed on campus -- a task force of the American Mathematical Society (AMS) spent 7 years interviewing the chairs of half the nation's Ph.D.-granting math departments and 30 of their deans. They found a dramatic dichotomy in perceptions. "There wasn't a dean who didn't say math was the leading cause of complaints" at his university, says AMS executive director John Ewing. And whereas most mathematicians saw their discipline as central to the sciences, says Ewing, most deans viewed mathematics as one of the most "insular" departments on campus. [9]

A lot of people have sensed that mathematicians, more than the rest of us, tend to live in their own heads. Recently some citable support for this view has appeared.

A recent study of three scientists with Asperger syndrome (AS) -- a mild version of autism -- suggests that deficiencies in "social" intelligence have no effect on math smarts....

Other recent research has indicated that autism is more common in families of physicists, engineers, and mathematicians.

[Richard] Borcherds [a recipient of the Fields Medal, math's equivalent of the Nobel Prize], now at the University of California, Berkeley, is frank about his condition, although he describes himself as being "at the fuzzy borderline" of Asperger syndrome. He's not sure the research says anything new. Mathematicians' social ineptness has long been part of the profession's self-deprecating folklore, he observes: "I seem to have a hell of a lot of colleagues who are not too much unlike me." [10]

Given that the reported "insularity" may even have a genetic foundation, the Mathsemantic Monitor finds it particularly fascinating that he's heard not only from math buffs but also from quite a few math educators. From Rhode Island to Australia, the educators he's heard from seem to have taken the book Mathsemantics [11] seriously, even though it stresses right from page one the need to blend math with the meanings of ordinary language. Chapter two opens with a complaint about the "schismatic instruction" of English and math. Chapter seven makes the point that math enthusiasts need to watch their language. Chapter nine, titled "Divorce," notes how students' math difficulties relate to ordinary language. Chapter ten makes the point that math alone is inadequate. And so on until chapter 24 argues the need for mathsemantic sophistication, laments its lack, and then discusses solutions (emphasis added).

I don't think we dare wait for institutionalized education to evolve enough to teach mathsemantic sophistication. In the normal course of events, we'd need to allow perhaps twenty years for educators to agree on the importance of mathsemantics, perhaps another ten to decide whether it should be taught by math teachers or English teachers, another twenty to train sufficient teachers, another twenty for them to train the first generation of mathsemantically sophisticated future leaders, and then perhaps another twenty-five for that full generation to reach actual leadership. Thus, after ninety-five years we could begin. That really won't do, will it?

The Mathsemantic Monitor admits, with the benefit of six years of hindsight (only eighty-nine years to go, then?), that he could have said still more plainly what he's tried to say in this current piece: For English-speaking people, mathsemantics requires a blending of English and math. Neither English nor Math savvy can do it alone.

Therefore, he'd like to amend the list of mathsemantic propositions. In the book these end with number 29.

29. Math without meaning could mean death by the numbers.

He'd now like to add a proposition 30 as a kind of plain summary.

30. Mathsemantic competence requires a reasonable combination of simple math and ordinary English (or whatever ordinary language you speak).

At this point the Mathsemantic Monitor would love to hear from some English buffs or from some English educators, say at least one, okay? Haven't any of you read my book? How about it? Let's start tearing down the barrier.

(*.) A persona of aviation-consultant, demalogician, etc., Edward MacNeal, a regular contributor to these pages.

NOTES AND REFERENCES

(1.) Norton Juster, The Phantom Tollbooth, 35th anniversary edition (New York: Random House, 1996).

(2.) Roxanne Nelson, "Prevented Prevention," in Scientific American, May 1999.

(3.) The Howell Book of Dog Care as quoted in Harrisburg Magazine, March 1999.

(4.) W. Wayt Gibbs, "The Magnetic Attraction," in Scientific American, May 1999. Analogous usages ("11-fold larger" and "42-fold less") appear in Dmitri Petrov, et al, "Evidence for DNA Loss as a Determinant of Genome Size" in Science, vol. 287, no. 5455 (February 11, 2000), and ("five times more children") in Karla Zadnik, et al, "Myopia and ambient night-time lighting," Nature, vol. 404, no. 6774 (March 9, 2000).

(5.) Presentation by the Mathsemantics Monitor at the Agnes Irwin School, April 10, 1997.

(6.) Personal e-mail communication to Dr. Peter Brinkworth, May 27, 2000.

(7.) Elizabeth Ruksznis, "Interdisciplinarity: Psychology + X," in Observer, American Psychological Society, vol. 12, no. 3 (March 1999).

(8.) Robert Service, "Exploring the Systems of Life," Science, vol. 284, no. 5411 (April 2, 1999).

(9.) "ScienceScope," Science, vol. 285, no. 5429 (August 6, 1999).

(10.) Constance Holden, "Math and Asociality," Science, vol. 287, no. 5457 (February 25, 2000).

(11.) Edward MacNeal, Mathsemantics: Making Numbers Talk Sense (New York: Viking, 1994).