Towing the line of infinity

After several months of long walks and taxi rides, yesterday I finally worked up the courage to pick up the phone and have my malfunctioning automobile towed to a mechanic. (It turned out that only one phone call was required; I had feared that two or three would be necessary, and that I would characteristically place them in the wrong order.) The long wait (by which I mean both the months of procrastination and the hours I spent yesterday waiting for the guy to show up) did not go unrewarded.

Upon climbing into the truck (it was, of course, news to me that tow trucks also transport the owners of towed vehcles), I instinctively reached for the seat belt — beginning a search which was destined to be futile. In short order, the driver offered me a lesson on the folly of seat belts; apparently, wearing one increases the risk of death or injury in the event of a crash! (Evidently, yet another case of the government conspiring to kill as many people as possible.) Well, I thought, at least I won’t have gotten through this day without learning something.

But that was only the beginning. It was not long before the inevitable conversation began:

– “So, I guess you’re at the university?”

– “That’s right.”

– “What do you study?”

– “Mathematics,” I replied, bracing myself.  

– “Ma-the-ma-tics!” the driver exclaimed, drawing out each syllable as if to emphasize the esoteric nature of the subject. “Tell me, what do you do with a degree in ma-the-ma-tics, other than teach other misguided souls that there’s actually some use for it?”

(Oh boy, here we go…)

– “Well,” I said, “one can also study mathematics itself as a discipline — do research, you know.”

– “And to what purpose?” he immediately asked, with the clear implication that only certain purposes were acceptable.

– “Curiosity, I suppose.”

– “Curiosity, eh! Curiosity killed the cat!” (It was around this time that I noticed the Confederate flag sticker above the windshield — rather unusual in a region of the U.S. where, for example, a university cancels its classes on the occasion of Rosh Hashana.)

Before I could say anything, he quickly conceded:

– “Actually, I think curiosity’s an okay thing.” That was good news. I mean, maybe I’m willing to have my mind changed about mere life-and-death matters like seat belts, but if this fellow was going to question the value of curiosity, that would be going too far!

He continued:

– “I have a degree in English, you know…”

– “Well, it doesn’t seem to have stopped you!” I pointed out. (Would you have guessed that a degree in a subject as “impractical” as English could qualify one for an occupation as “practical” as that of tow-truck operator? Me neither.)

–”Yeah, well….I could never get interested in mathematics. Geometry and logic and such I could handle, but when it came to real mathematics… My father was always trying to get me interested in mathematics. I’ll tell you exactly when I realized that mathematics was useless. One day my father sat me down and said ‘I’m gonna show you something really interesting: that 0.999 to the n is equal to 1′. And he wrote out a proof that 0.9 to the n is equal to 1, and he said to me, ‘Isn’t that interesting?’ And I said, ‘I have a question’, and my father said ‘Okay’. So I said, ‘A decimal is a way of writing a fraction, right?’ And he said ‘yes’. And I said, ‘A fraction is by definition a part of the whole, right?’ And he said ‘yes’. So I asked ‘How can a fraction like 0.9 to the n be equal to 1, which represents the whole?’ And my father scowled and went away. And I realized that if mathematics allowed you to prove things that contradict logic, like a part being equal to the whole, then I had no use for it.”

(Obviously, by “0.9 to the n” the driver and/or his father must have meant , i.e. an infinite string of 9’s after the decimal point.)

– “Surely you don’t still hold to this view, do you?” I asked incredulously, referring to the idea that fractions are necessarily less than “the whole”.

– “Why yes, of course!”

– “Oh, well, it’s quite wrong,” I said, in that softly authoritative ivory-tower tone of voice. The driver smirked and snorted, as if to say, “here comes another one of these pointy-headed academics trying to tell me that things don’t mean what they mean!”

I asked:

– “What about 1.5? That’s a decimal — hence a fraction. But it’s greater than 1, which shows that a fraction can be not only equal to the whole, but even greater than the whole!”

– “No, no, that’s one, point five. That point five is zero point five, a fraction, less than one. Any time you have zero point some number, it’s always less than one. It doesn’t matter if you have a million nines, or a billion, or any number, it’s still less than one.”

– “No mathematician would argue with that!”

– “Then how come my father proved that 0.9 to the n is equal to 1?”

– “If you have infinitely many 9’s, that’s a different story — that means the 9’s never stop. If they stop somewhere, then it’s less than 1, of course. But if they never stop, then it’s equal to 1.”

– “That would defy logic! It doesn’t matter how many 9’s you put on; you’ll never get to 1.”

I was of course totally unprepared for this discussion, in addition to working under the stress of time pressure (by this time we had already arrived at our destination), so I wasn’t able to employ the obvious arithmetical demonstration of who was really “defying logic”. Nor did I remember to ask the driver the most important question: What, exactly, was his father’s argument that 0.99999….. should be equal to 1, and where was the flaw in his reasoning? The best I could do was:

– “So what you’re saying, basically, is that you don’t believe infinity exists — that it simply doesn’t mean anything to you to write infinitely many 9’s after the decimal point.”

– “What I’m saying is that you can put as many 9’s as you like — infinitely many or whatever — and it will always be less than 1.”

– “Okay, let me ask you this. Suppose you take this number 0.999… with infinitely many 9’s, which you say is less than 1, and subtract it from 1. What do you have left?”

– “Who knows? Some infinitely small number.”

I certainly didn’t have time to go into a discussion of infinity, and of Cantor and Kronecker, let alone Abraham Robinson and John Conway. It would have been complicated anyway, because the driver contradicted himself once we were outside the vehicle:

– “The problem is that mathematics and science and physics and geology and all that — these are finite subjects. Once we start introducing ideas of infinity into the picture, then we’re talking about philosophy. And philosophy should not be mixed into mathematics and science.”

I really had no idea where to even begin with this, so I just kept listening, while he unfastened my vehicle from the tow truck.

– “I guess what I’m saying is that I don’t want it to be true.” (Always a knockdown argument!). “If all the sudden 5.9 were equal to 6 and so on, my whole world would be out of whack — buildings that were supposed to be straight would be slanted…”

And then…

– “It’s like religion, organized religion. One of the biggest is Catholicism, you know, Roman Catholicism. They used to say that it was a sin to eat meat on Fridays. So when I was young the schools couldn’t serve meat on Fridays. And then all of a sudden they changed their minds and said it was okay after all. Now if something could go from being a mortal sin to being perfectly okay on a dime, just because somebody said so, then that’s a way of thinking I don’t have any use for.”

I don’t happen to know the details (historical or current) of the Catholic Church’s policy on meat-eating on Fridays or any other day, but at any rate, here at last was a principle that this Confederate-flag-displaying, 0.999…= 1 denying seatbelt skeptic and I could agree on!

When I wrote out a check to this guy, I was careful to write “60.00″ and not “59.9999…”.