Following is an excerpt from
J.A.Paulos,
Beyond Numeracy

# A MATHEMATICAL ACCENT oo

The practical importance of studying mathematics is widely acknowledged, but relatively few will grant that the mathematics of everyday life can be an engaging topic for idle thought. Mathematics, however, provides a way of viewing the world, and developing a mathematical consciousness or outlook can enhance our daily rounds.

Rather than argue for this, let me illustrate with an extended anecdote. Recently I had to drive to New York on short notice and I was a little late. Behind a line of cars on the way to the turnpike, I developed the usual murderous thoughts as I noted that the driver in the lead car was letting cars from a busy side street turn right onto the street before him (and me). There was a traffic light at the intersection, so there was no need for this bit of philanthropy, and the would-be samaritan should have balanced his good deed against the aggravation to the drivers behind him. The mathematical integral or sum of the latter inconveniences was in this case greater. Although not a deep thought by any means, it and similar "calculations" seem to be quite foreign to many.

Finally reaching the expressway, I quickly accelerated to the prevailing speed of approximately 70 miles per hour, slowing down to something near 55 only when patrol cars appeared. Despite my rush, the inanity of the game seemed especially stark that day, and I wondered why no one had ever implemented the following simpleminded idea for curbing excessive speed on toll roads: When someone enters such a road, the ticket he or she receives is stamped with the time of entrance. The distance between the various toll booths is known, so that when the exit time is recorded later by a computer, it can easily calculate the person's average velocity while on the toll road. The toll booth operator may then direct speeders with incriminating toll tickets to a waiting patrol car.

The method wouldn't do away with all speeding, of course, since someone could still speed until just before the exit, stop for a cup of coffee or a full meal if they'd really been racing, and exit with a legal average velocity. Still the primary inducement for speeding would be gone. What's wrong with this plan? Division of one number by another, the distance traveled by the time elapsed, is surely not a risky or a novel technology. Speeding tickets are presently issued on the basis of radar, which is considerably less reliable.

Turning on the radio to escape the topic, I was reminded how I would so like to hear just once a rock song that uses the word "doesn't" rather than "don't" as in "She don't love me anymore" or, what was actually playing, "It don't matter anyway." Possibly because of the relative sensory deprivation of driving, the latter lugubrious lyric stuck with me. Maybe it didn't matter anyway, and if so, I wondered if it mattered that it didn't matter. If nothing mattered and that nothing mattered didn't matter either, then why couldn't we iterate? It didn't matter that it didn't matter that nothing mattered. And so on recursively.

I inhaled the fumes of the New Jersey Turnpike and further considered the situation. If nothing mattered, but that nothing mattered did matter, then we would have a rather dispiriting situation. If nothing mattered and that nothing mattered didn't matter either, then we would have the possibility of something better-an ironic and conceivably happy approach to life. Similarly at higher levels. Reasoning formally and probably simplistically, the best situation would be for things to matter at the basic level or, failing that, to matter on no level either complete childhood simplicity or total adult irony. (See the entry on time.)

As I neared the Hess oil refinery, my thoughts turned to writing and publishing, but my absurdist mood persisted. Was there, given the large and increasingly homogeneous reading population, less of a "need" today for writers? Assuming that people read about the same number of books; magazines, and newspapers as they always did, and that they wanted to read the "best" of whatever they did read (as determined by best-seller lists, say), and that they tended, in large part, to read material written by their countrymen, it seemed to follow that the larger the nation, the smaller the percentage of its citizens who could be authors or, what one day might become equivalent, best-selling authors.

I thought of several counter-arguments, the most interesting pointing to the larger variety of publications (especially nonfiction books, magazines, and newsletters) that catered to ever more specialized tastes and that provided more opportunities for writers. If there was any substance to these vague musings, the likelihood of attaining literary stardom was shrinking, while the chances of earning a living with one's word processor were rising.

The radio was reporting a one-hour delay in the Lincoln Tunnel, so I decided to take the George Washington Bridge into Manhattan. This turned out to be not much better, since the victims of a minor accident were on the side of the road attracting the usual rubberneck response from passing motorists. The cumulative effect of everyone's slowing down to see that there was really nothing to see struck me as a miniaturized version of many human problems. No evil intent, just a common impulse whose magnification was unpleasant.

Traffic cleared after a twenty-minute delay only to clot up again even worse due to construction. A single-lane stretch of about a mile before the bridge was needlessly peppered with Do Not Pass signs. The signs reminded me of progressive sentences in which each succeeding word is one letter longer than its predecessor, and I whiled away the time dilating on the pattern. Finally I came up with "I Do Not Pass Since Danger Expands Anywhere Unmindful Speedsters Proliferate Unmanageably," of which I was inordinately proud.

Tiring of this, I noted the increasing frequency of MD license plates as we approached New York and remembered the statistic I had just read that there are 428 physicians in all of Ethiopia-a country of 40,000,000 people whose average life expectancy was under 40. Trying to keep my impatience at bay, I constructed biographies of people from their vanity license plates and concluded without a shred of evidence that I was right every time. This brought to mind the punch line of a license-plate joke told me by a mathematician friend: How do you spell the name "Henry"? Answer: H-E-N-3-R-Y-the 3 is silent.

On the bridge I remembered that the supporting cables take the form of a curve called a catenary unless weights are hung from them at equal intervals, in which case the shape is parabolic. I considered the likelihood of the bridge's collapse and then the still unlikely, but vastly more probable possibility of being killed by a drunk driver, or ultimately contracting cancer from my repeated exposure to the New Jersey Turnpike, or developing high blood pressure from the frustration of being locked in a car alone with my obsessive ruminations.

I arrived in New York only five minutes late for my appointment, but that isn't the point of the story. Its purpose was to illustrate a mathematical stream of consciousness. The topics which arose naturally were social trade-offs (the good samaritan, gawking at accidents), average velocity (speeding tickets on the turnpike), the logical level of statements (the "nothing matters" business), probability (chances of becoming a published author), wordplay (snowball sentences), and estimation (dying from a bridge collapse vs. other, more likely demises).

For most nonscientists, what's most important in science education is not the imparting of any particular set of facts (although I don't mean to denigrate factual knowledge), but the development of a scientific habit of mind: How would I test that? What's the evidence for it? How does this relate to other facts and principles? The same, I think, holds true in mathematics education. Remembering this formula or that theorem is less important for most people than is the ability to look at a situation quantitatively, to note logical, probabilistic, and spatial relationships, and to muse mathematically. (See the entry on computation and rote.)

Of course, I'm not advocating an exclusive focus on such musings, just a realization that mathematics is much more than computation, that the outlook that results from studying it can illuminate aspects of our lives other than our financial or scientific concerns. At the very least it can provide an alternative way to fill our driving time.