A few days later I was on my way to Physics when I bumped into Stoney. Odd. I’d never bumped into him before, except in class. He was wearing bell-bottomed Levi’s, his Durango boots, a puffy-sleeved shirt that looked like it might have been intended for one of the pirates of Penzance, and his sunglasses.
“Whoah,” he said. “Fancy meeting you here.”
“I’m surprised to see you, too. For some reason I thought you weren’t an early riser.”
“Yea, well, actually, I’m not. For me this is more of a late night than an early morning. What are you up so early for?”
“Stoney, it’s 10:00.” I said.
“Yes?”
“Not the crack of dawn,” I said. He thought for a few seconds.
“So I take it you’re functional at this time of day with some frequency?” he asked.
“On my way to Physics. Happens three times a week.”
“Far out. Have fun. Physics. Man,” he said. He shook his head, but seemed happy for me.
“You want to come?”
“To a class?” he asked, dumbfounded.
“Sure. Physics. You might find it interesting. Plus, you’d almost certainly meet Rob and Toni.” He thought for a few seconds, as though confused.
“And this would interest me in some way?” he asked.
“They’re both completely insane, but smart. She’s gorgeous,” I said.
“Man, I don’t go to classes even when I’m registered for them. Suggesting I might go to one I’m not signed up for seems pretty fucked up.”
“Are you sober?” I asked. He thought about his answer before speaking.
“’Sober’ would be inaccurate. In any sense of the word. But ‘sober’ generally refers to one’s interaction specifically with alcohol, and alcohol is not the largest sector of my current weltanschauung, although it is well-represented.”
“How can you be fucked up at 10:00 a.m.?” I asked.
“Um, like I said, this is more of a late night than an early morning.”
“Did you take Physics at Lawrenceville?”
“Sure. It was a little silly,” said Stoney.
“How so?”
“All these extremely precise calculations that don’t relate to much reality.”
“How so?” I asked.
“Math doesn’t describe reality. Reality is, like, reality. Physics is a bunch of guys standing around looking at cannon balls dropping from towers for a couple of hundred years, and eventually one of them invents a clock, then another one figures out how to put some numbers on a page that predict when it’s going to hit the ground. Later somebody calls it acceleration and gets famous for figuring things about it. Then a hundred years later some other guy comes along and figures out a way to do the calculation that’s a little more accurate and he gets famous for that. It’s fucked up.” He paused and fumbled to find a cigarette, eventually found a soft pack of Marlboros somewhere in his boot, extracted one, and lit it with a disposable lighter. He looked up at me after taking the kid of puff a person who doesn’t smoke all the time takes.
“What were you saying?” he asked.
“You were explaining how Physics is messed up because people keep coming up with more precise formulæ.”
“Oh, fuck yes. And every now and then somebody comes along and has a completely different way of looking at the problem of “what is acceleration?” and you Physics guys all look at things differently. And then everybody in all the schools has to teach the new deal and everybody coalesces around Einstein. Or Bohr. Or Copernicus. Or who the fuck cares. You should read Kuhn. What?” I was looking at him and waiting for him to finish, and he was reacting to my expression.
“Nothing. I just get this a lot,” I said.
“What?”
“Math people telling me that physics is imprecise.”
“It fucking is imprecise. Has to be,” he said.
“And nothing like this happens in Math?” I asked.
“Oh, fuck no. Nobody ever came along and proved that 2+2=5. Nobody ever will. Nobody ever, since time began, came along and showed all the other mathematicians that the times tables were wrong, or that addition isn’t associative. Math just isn’t like that. Once something is so, it’s so for all time. Math just is.”
“I gotta go. Want to come to class with me?” I asked.
“What’s class about?”
“Kinematics. Dynamics. Mechanics.” He thought a few seconds and took another non-smoker’s puff.
“I wanna give you shit for talking like a Physics guy, but I took high school Physics so I know what that all means”
“So?”
“I say we blow it off and smoke some pot instead,” he said.
“Don’t smoke,” I said.
“Oh, right.” He thought for a few seconds. “Okay I’m game. Lead on, MacDuff, ” he said.
“Cool,” I said. “Your presence is going to surprise Toni and Rob.” I turned and headed for the Science and Math Center, and Stoney came along with me.
“Who are they?”
“They’re crazy people who sit on either side of me.”
“If it’s assigned seating, where are you going to put me?” he asked.
“No, it’s not assigned seating,” I said.
“So why do Rob and Toni always sit next to you?”
“They’re fucked up.”
“Like, stoned?” he asked.
“No. Like utterly insane,” I answered.
“I’m not completely stoned,” he said. “I should be able to deal. She’s the good-looking one, right?”
“Yes, but wacko like you don’t often encounter.”
“I’m not sure a person who doesn’t do drugs is qualified to make a statement like that,” he said. We were still walking, more slowly than I usually would, towards the lecture hall where Physics met.
“Okay,” I said.
“Like, once, when I was on acid, I was sure a red dragon with yellow dorsal plates and ivory teeth was licking the inside of my nose.”
“Yuck,” I said..
“It actually felt pretty good,” said Stoney, reflectively. “She, or maybe he, for that matter, what do I know about dragons? had a very soft and narrow tongue. Warm. It moved around in ... interesting ways.”
“Odd,” I said. There was a brief reverie in which Stoney seemed at a loss for words. “Did you reciprocate?” I asked.
“How do you mean?” he asked.
“Did you lick the red dragon’s nose?”
“Are you fucking nuts? Have you ever seen a dragon’s nose?” he asked.
“Um, no,” I answered, patiently, pretty sure that Stoney hadn’t seen one, either.
“Their noses are just like pigs’. Runny and dirty and noisy. No way I’m licking a dragon’s nose. That’s just gross.”
We had reached the amphitheater at the Science and Math Center. We were slightly, but not very, early as we moved into the amphitheater. I took a seat in the middle of the hall, and Stony sat to my right.
“So why does Physics have to be taught in such a big room? A kind of a … steep room,” Stoney asked.
“It doesn’t, but this is a course all physics majors have to take, so it’s a big class, and this is a big classroom.”
“The physics guys and gals are actually pretty cute, now that I look at them,” he said.
Right as he said that, Rob showed up on my left and Toni showed up on Stoney’s right.
“Excuse me, you’re in my seat,” Toni said to Stoney. Stoney, still in his sunglasses, looked up, confused.
“Excuse me?” he asked.
“You’re in my seat,” she said. I stood.
“Oh, right,” said Stoney. “You’re the one Henry told me about.”
“Look, Stoney, she’s eccentric. This will go a little smoother if you sit on my left instead of my right.”
“What did he tell you about me?” Toni asked, a little more intensely than comfort allowed.
“Well,” Stoney said, noncommittally, glancing at me.
“Feel free to be honest,” I said.
“He said you were gorgeous but … a little … eccentric.”
“Henry?” she asked.
“I said you were beautiful and brilliant but crazy,” I said. “Stoney, move to this seat,” I said, gesturing with my left hand. It will be a little easier to deal with Rob.”
“Is this more of your sexist patronizing bullshit?” she asked me, but addressing the world in general.
“Yes,” I answered. “Stoney, sit over here.” Stoney stood and moved past me to sit in the seat to my left. Toni sat down with too much ado. Students around us were watching and laughing. Toni and Rob and I were dependable entertainment, and Stoney was a new addition to the act.
As Stoney passed me to move to the other seat, he paused for a second. “Her nipples got hard while she was yelling at us,” he whispered. “That’s so cool.”
“I heard that,” she said. Stoney shrugged.
“Hi,” said Rob, sticking out his hand towards Stoney. “I’m Rob.” He’d been standing there watching us, silently and patiently Rob-like, while we dealt with his girlfriend.
Stoney said “Yo. Rob. I’m Stoney,” and stuck out his hand. Rob looked at it, confused, then grasped it for a second.
“Is there a problem?” Rob asked, still standing.
“No. Rob. This is my friend Thomas Jackson. He, like me, is a math major.”
“You were a Physics major last time we talked,” said Rob.
“I’m a double major. Thomas—” I started.
“You can call me Stoney,” he said.
“Thanks,” said Rob and shook his hand again.
“Thomas—” I started.
“Stoney,” both Rob and Stoney corrected me, in unison.
“Stoney’s a friend of mine from the Math Department, so I invited him to class just to see what Physics is like. So I know you like being close to Toni, but if this once you could sit one seat further to the left, we could keep Stoney up to date on what’s going on. Rob looked at Stoney carefully.
“Did you take high school physics?” Rob asked.
“Sure,” he answered.
“What’s Newton’s second law?”
“Acceleration is proportional to force and inversely proportional to mass,” Stoney answered.
“Okay,” said Rob, although he didn’t sit down. He was still looking for a reason not to go along with the new seating scheme, Stoney shrugged and sat down between us.
“Henry gets very good grades bit he didn’t know the answer to that question last class, so far as I could tell,” said Rob, still standing. “Prof. Dannhausen doesn’t generally state first principles,” Rob said, erroneously.
“That is a good way of expressing the last set of problems,” I said, to Rob. I had understood Newton’s second law perfectly well but absolutely nothing will get me to engage in an argument with Rob.
“I’m just glad you didn’t ask me about the second law of thermodynamics ,” Stoney said. “That fucker’s fucked.”
“What?” said Rob and Toni, simultaneously.
“Yeah, it just is,” he answered.
“Why?” they demanded, simultaneously. Stoney was put off by their keen attention.
“Okay, well, it’s true that if a high energy system comes in contact with a low energy system, energy will flow from the higher to the other. To the lower.” Stoney looked up. He now had the attention of the students in the next row back. “Take a hot marble and a cold marble, put ‘em together, pretty soon they’re the same temperature,” he said. Everyone nodded.
“But look at what happens in the universe. A star explodes and creates new elements. Gases cool and form solar systems and shit A planet cools, life emerges and crap, then evolves into fuckers like us. In thermodynamics, you physics fuckers act like each step cooler is a step towards annihilation, because we’re just a little bit closer to absolute zero. But each step was more highly ordered. Stars. Elements. Life. Not much order a second after the Big Bang. Just a bunch of hot quarks and shit. Now we have sex and basketball and mushrooms. Far more organized.” Everyone listening applauded. Stoney stood and curtseyed as though wearing a short skirt.
“Hi, everybody,” I said. “This is my friend Thomas Jackson. He’s auditing this class today. He’s smart.”
“Call me Stoney. And I’m not as much smart as I am totally stoned,” he said. At this point the door right of the lectern swung open suddenly and Professor Dannhausen strode in with a flourish.
“Rob, look. Just for today, can you sit just one seat over? I know you like being closer to Toni, but I promise you she’ll still love you if you let Stoney sit next to me today.”
“I never said I loved Rob,” said Toni, from my right.
“I promise you she won’t have sex with you any less if you just sit down,” I said. Rob sat down in the seat next to Stoney without comment. And I sat between Stoney and Toni.
“Don’t get pissed at him about this, Toni. This is my deal, and he’s doing me a favor.” We had, by this point, attracted Dr. Dannhausen’s attention.
“Mr. Baida, is class ready to begin?” Dr. Dannhausen asked me. He didn’t overstress anything. He wasn’t an asshole, he was just accustomed to starting his class on time.
“Yes, sir. Sorry about the delay.”
“And you have a new and colorfully attired student next to you?” said Dannhausen.
“Yes. This is my friend Thomas Jackson.”
“Yo. Call me Stoney, prof,” he said. Dannhausen cocked an eyebrow at me.
“He’s a Math major. It seems to me like there’s a lot of overlap between the two disciplines, and I thought he might be interested. He promises not to cause trouble,” I said.
“Absolutely. I’m a laid back, cool kind of mathematician,” said Stoney. He still hadn’t taken off his sunglasses.
“Okay. Welcome, Mr. Jackson. So, Mr. Baida, how about you explain to me and to Mr. Jackson what Newton’s third law is.”
“Fuck. I totally know this,” said Stoney.
“Hush, Stoney,” I said, He put his hand to his mouth in embarrassment. “For every action there is an equal and opposite reaction,” I said.
“Right. Give me an example.”
“When I push on a brick, the brick pushes back.”
“Right. So let’s do a problem. Assume I have two canoes on a river, one with two occupants with a combined mass of 150 kilograms, and the other with three occupants with a combined mass of 250 kilos. The occupants push away from each other with a combined force of 43 newtons.”
“Weaklings,” said Stoney,” under his breath.
“How do we find the acceleration of the two canoes?” Prof. Dannhausen asked me. He didn’t usually focus on one student like this. He usually stated off by describing a principle, then explaining the basic concepts of the calculations, and then gave us a few calculations. He was zooming straight through to the end. In his defense, though, Newton’s third law is pretty basic. There wasn’t any way to build dramatic tension about it.
“Well, according to the text, the acceleration is described as Σ Fx = max for ax.”
“According to the text?” he asked.
“Yes, sir,” I said.
“Do you not believe the text?”
“Yes. Sort of. I’m not sure. The text never comes out and says these formulas are approximations, but it never comes out and says somebody’s sat down and tested them a thousand times and they were 100% accurate, either. I’m just not sure the formula is more than a description of what we see.” Stoney was smiling broadly, but all of the sudden I was back in Mr. Finch’s Sunday school class and I was challenging orthodoxy and realized I was about to get verbally slapped. The other kids were pulling back from me, and I braced for it. There was a dramatic pause.
“Interesting, and possibly right,” said Prof. Dannhausen. “But let’s not get sidetracked. We’re talking about Newton’s second law and five people in two canoes. Let’s work through the problem.” No slap. Weird.
We talked through the problem. If you’re curious the second canoe is (and we’re solving the second one first because it has the greater mass):
ax = Σ F2x = 46 N = 0.18 m/s2
m2 250 kg
and the first canoe is
ax = Σ F1x = -46 N = -0.31 m/s2
m1 150 kg
and the reason this problem exemplifies Newton’s third law is that the force, 46 newtons, is applied to the formulæ for both canoes—positive 46 newtons in the top equations (look for 46N in the numerator) and negative 46 newtons in the bottom. It took a while to work through it. So you’ll notice that the same force applied to the lighter canoe caused it to accelerate more than the heavier canoe, because the heavier canoe is going in one direction at 0.18 meters per second, and the lighter one is going in the other direction at not quite twice as fast, or 0.31 meters per second. It took most of the class to go through the calculations, for acceleration, rate, and distance, and to discuss the implications of each step, but at the end, the canoes were 0.35 meters apart.
“So is this a realistic calculation?” he asked the class.
“Oh, fuck,” said Stoney.
“No,” said Toni, to my right, loudly.
“Yes, Ms Sayers?”
“The solution ignored friction.”
“Exactly,” said prof Dannhausen.
“Lord fuck a duck,” said Stoney, in what he might have thought was a quiet voice. Prof. Dannhausen cocked an eyebrow at him.
“You think something else was left out of the solution, Mr. Jackson?” Prof. Dannhausen asked him.
“Well, yeah,” said Stoney. “You said the canoes were in a river. So there’s going to be current, and there’s no way two different canoes of different masses would react the same to it. And canoes are really unstable, and they wouldn’t react to being pushed like that. They wouldn’t move in a straight line. This whole point mass concentration thing is a fiction. Kind of. I mean, I know I’m a guest and all, and I’ve enjoyed watching you and my gay friend Henry solve the canoe deal, but it doesn’t really describe what would happen, does it?” Prof. Dannhausen put down his chalk and made a face that communicated “I wasn’t expecting this, but okay.” Another pause while he thought.
“Our visitor is largely correct. If you understood, please raise your hand.” Rob, Stoney, I, and Toni raised our hands, as did about a quarter of the class.
“First,” Prof. said, “Most of the problems you will encounter this year, as all of the problems you encountered in high school, assume that all of the masses involved are concentrated into points. Newtonian mechanics almost always makes this assumption, from the simplest vectors to planetary mechanics. Newton knew this assumption was inaccurate, and seems to have figured out more accurate calculations as to planetary motion, but refused to have them published because the Church was unhappy with his forst set of calculations, which agreed with Galileo and Copernicus. The point mass concentration idea is a sort of convention. The mass in a canoe containing two live bodies is most certainly not concentrated into a point. The center of gravity would be difficult to calculate, people move, changing things, if they spin, the centrifugal and centripetal forces would change. The canoe would react to the current. All of this would be …”
“Turbulent,” said Toni.
“Chaotic,” said Stoney.
“Both good descriptors. What I’m teaching you here is the basics of Newtonian dynamics and mechanics. Stripped to its bare bones. So the way Mr. Baida and I worked through the canoe problem was not realistic, and if you choose to study this area in greater depth you will learn that mechanics is far more complicated than we are teaching here. But you need to understand these basics before we can develop the complexities. Forgive me if our problems are oversimplifications, but they are necessary simplifications.” He paused and looked at the floor. “I also want to address something Mr. Baida said earlier. I believe that the equations and formulas you’re learning are accurate to a very precise degree. But no one can tell you that they’re true, because the fact that they describe the universe very accurately does not mean that there can’t be a different formula out there that describes things more accurately. It’s happened many times in physics that we thought we knew what we were looking at, then someone came along and showed us we were wrong. But, I’m not teaching you anything I know to be untrue.” He looked at his watch. “I think that’s all for today,” he said. “Mr. Jackson, come back any time.”
“Yeah, thanks,” he waved, and we all started standing up. “And none of you guys are stoned for this shit?” asked Stoney, to anyone in earshot.
“You’re gay?” Rob asked me.
“No.”
“Oh, he is, totally,” said Stoney. “This Peabody girl named Ginny was all over him and he didn’t even notice.”
“Stoney, give it a rest.”
“Henry, is this true?” asked Toni, a little aggressively.
“No, Toni, it’s not true.”
“Some people just can’t accept themselves for who they are,” Stoney said, as we walked out.
Sunday, May 23, 2010
Thursday, May 20, 2010
Chapter 20: A Chance Encounter with Stoney; Note that You Get the Joke Even if You Don't Do Math
A few days later I was on my way to Physics when I bumped into Stoney. Odd. I’d never bumped into him before, except in class. He was wearing bell-bottomed Levi’s, his Durango boots, a puffy-sleeved shirt that looked like it might have been intended for one of the pirates in The Pirates of Penzance, and his sunglasses.
“Whoah,” he said. “Fancy meeting you here.”
“I’m surprised to see you, too. For some reason I thought you weren’t an early riser.”
“Yea, well, actually, I’m not. For me this is more of a late night than an early morning. What are you up so early for?”
“Stoney, it’s 10:00.” I said.
“Yes?”
“Not the crack of dawn,” I said. He thought for a few seconds.
“So I take it you’re functional at this time of day with some frequency?” he asked.
“On my way to Physics. Happens three times a week.”
“Far out. Have fun, Physics. Man,” he said. He seemed happy for me.
“You want to come?”
“To a class?” he asked, dumbfounded.
“Sure. Physics. You might find it interesting. Plus, you’d almost certainly meet Rob and Toni.” He thought for a few seconds, as though confused.
“And this would interest me in some way?” he asked.
“They’re both completely insane, but smart. She’s gorgeous,” I said.
“Man, I don’t go to classes even when I’m registered for them. Suggesting I might go to one I’m not signed up for seems pretty fucked up.”
“Are you sober?” I asked. He thought about his answer before answering.
“’Sober’ would be inaccurate. In any sense of the word. But ‘sober’ generally refers to one’s interaction specifically with alcohol, and alcohol is not the largest component of my current weltanschauung.”
“How can you be fucked up at 10:00 a.m.?” I asked.
“Um, like I said, this is more of a late night than an early morning.”
“Did you take Physics at Lawrenceville?”
“Sure. It was a little silly,” said Stoney.
“How so?”
“All these extremely precise calculations that don’t relate to much reality.”
“How so?” I asked.
“Math doesn’t describe reality. Reality is, like reality. You get a bunch of guys standing around looking at cannon balls dropping from towers for a couple of hundred years, and eventually one of them invents a clock, then another one figures out how to put some numbers on a page that predict when it’s going to hit the ground. Later somebody calls it acceleration and gets famous for figuring out things about it. Then a hundred years later some other guy comes along and figures out a way to do the calculation that’s a little more accurate and he gets famous for that. It’s fucked up.” He paused and fumbled to find a cigarette, eventually found a soft pack of Marlboros somewhere in his boot, extracted one, and lit it with a disposable lighter. He looked up at me after taking the kid of puff a person who doesn’t smoke all the time takes.
“What were you saying?” he asked.
“You were explaining how Physics is messed up because people keep coming up with more precise formulae.”
“Oh, fuck yes. And every now and then somebody comes along and has a completely different way of looking at the problem of “what is acceleration?” and you guys all look at things differently. And then everybody in all the schools has to teach the new deal and everybody coalesces around Einstein. Or Bohr. Or who the fuck cares. You should read Kuhn. What?” I was looking at him and waiting for him to finish, and he was reacting to my expression.
“Nothing. I just get this a lot,” I said.
“What?”
“Math people telling me that physics is imprecise.”
“It fucking is imprecise. Has to be,” he said.
“And nothing like this happens in Math?” I asked.
“Oh, fuck no. Nobody ever came along and proved that 2+2=5. Nobody ever will. Nobody ever, since time began, came along and showed all the other mathematicians that the times tables were wrong, or that addition isn’t associative. Math just isn’t like that. Once something is so, it’s so for all time. Math just is.”
“I gotta go. Wanna come to class with me?” I asked.
“What’s class about?”
“Kinematics. Dynamics. Mechanics.” He thought a few seconds.
“I want to give you shit for talking like a physics guy, but I took high school physics so I know what all that means”
“So?”
“I say we blow it off and smoke some pot instead,” he said.
“Don’t smoke,” I said.
“Oh, right.” He thought for a few seconds. “Okay I’m game,” he said.
“Cool,” I said. “Your presence is going to surprise Toni and Rob.”
“Who are they?”
“They’re crazy people who sit on either side of me.”
“If it’s assigned seating, where are you going to put me?” he asked.
“No, it’s not assigned seating,” I said.
“So why do Rob and Toni always sit next to you?”
“They’re fucked up.”
“Like, stoned?” he asked.
“No. Like utterly insane,” I answered.
“I’m not completely stoned,” he said. “I should be able to deal. She’s the good-looking one, right?”
“Yes, but wacko like you don’t often encounter.”
“I’m not sure a person who doesn’t do drugs is qualified to make a statement like that,” he said. We were walking, more slowly than I usually do, towards the circular lecture hall where Physics met.
“Okay,” I answered.
“Like, once, when I was on acid, I was sure a red dragon with yellow dorsal plates and ivory teeth was licking the inside of my nose.”
“Ick,” I said, quoting Mary.
“It actually felt pretty good,” said Stoney, reflectively. “She, or maybe he, for that matter, what do I know about dragons? had a very soft and narrow tongue. Warm. It moved around in ... interesting ways.”
“Okay.” There was a brief reverie in which Stoney seemed at a loss for words. “Did you reciprocate?” I asked. He’d seemed pretty into it.
“How so?” he asked.
“Did you lick the red dragon’s nose?”
“Are you fucking nuts? Have you ever seen a dragon’s nose?” he asked.
“Um, no,” I answered, patiently, pretty sure that Stoney hadn’t seen one, either.
“Their noses are just like pigs’. Runny and dirty and noisy. No way I’m licking a dragon’s nose. That’s just gross.”
We had reached the amphitheater at the Science and Math Center. We were slightly, but not very, early. I took a seat in the middle of the hall, and Stony sat next to me.
“So why does Physics have to be taught in such a big room? A kind of a steep room.”
“It doesn’t, but this is a course all physics majors have to take, so it’s a big class, and this is a big classroom.”
“The physics guys and gals are actually pretty cute, now that I look at them,” he said, sitting down.
Right as he sat, Rob showed up on my left and Toni showed up on Stoney’s right.
“Excuse me, you’re in my seat,” Toni said to Stoney. Stoney, still in his sunglasses, looked up, confused.
“Excuse me?” he asked.
“You’re in my seat,” she said.
“Oh, right,” said Stoney. “You’re the one Henry told me about.”
“Look, Stoney, she’s eccentric. It will all go a lot easier if you sit on my left instead of my right.”
“What did he tell you about me?” Toni asked, a little more intensely than comfort allowed.
“Well,” Stoney said, noncommittally, looking at me.
“Feel free to be honest,” I said.
“He said you were gorgeous but … a little … eccentric.”
“Henry?” she asked.
“I said you were beautiful and brilliant but crazy,” I said. “Stoney, move to this seat,” I said, gesturing with my left hand. “It will be a little easier to deal with Rob.”
“Is this more of your sexist patronizing bullshit?” she asked.
“Yes,” I answered. “Stoney, sit over here.” Stoney stood and moved past me. Toni immediately sat down with too much ado. Students around us were unabashedly watching and laughing. Toni and Rob and I were dependable entertainment, and Stoney was a new addition to the act.
As Stoney passed me to sit on the other side, he paused for a second. “Her nipples got hard while she was yelling at us,” he whispered. “That’s so cool.”
“I heard that,” she said. Stoney shrugged.
“Hi,” said Rob, sticking out his hand towards Stoney. “I’m Rob.” He’d been standing there watching us, silently and patiently Rob-like, while we dealt with his girlfriend.
Stoney said “Yo. Rob. I’m Stoney,” and stuck out his hand. Rob looked at it, confused, then grasped it for a second.
“Is there a problem?” Rob asked, still standing.
“No. Rob. This is my friend Thomas Jackson. He, like me, is a math major.”
“You were a Physics major last time we talked,” said Rob.
“I’m a double major. Thomas—” I started.
“You can call me Stoney,” he said.
“Thanks,” said Rob and shook his hand again.
“Thomas--” I started.
“Stoney,” both Rob and Stoney corrected me, in unison.
“Stoney is a friend of mine from the Math department, so I invited him to class just to see what college Physics is like. So I know you like being close to Toni, but if this once you could sit one seat further to the left, we could keep Stoney up to date on what’s going on. Rob looked at him carefully.
“Did you take high school physics?” Rob asked.
“Sure,” he answered.
“What’s Newton’s second law?”
“Acceleration is proportional to force and inversely proportional to mass,” Stoney answered.
“Okay,” said Rob, although he didn’t sit down. He was still looking for a reason not to go along with the new seating scheme, Stoney shrugged and sat down between us.
“Henry gets very good grades bit he didn’t know the answer to that question last class, so far as I could tell,” said Rob, still standing. “Prof. Dannhausen doesn’t generally state first principles,” Rob said.
“That is a much simpler way of expressing the last set of problems,” I said, to Rob. I had, of course, been aware of Newton's laws last class, but there's no way I was getting in an argument with Rob, regardless of the topic.
“I’m just glad you didn’t ask me about the second law of thermodynamics ,” Stoney said. “That fucker’s fucked.”
“What?” said Rob and Toni, simultaneously.
“Yeah, it just is,” he answered.
“Why?” they demanded, simultaneously. Stoney was put off by their keen attention.
“Okay, well, it’s true that if a high energy system comes in contact with a low energy system, energy will flow from the higher to the other. To the lower.” Stoney looked up. He now had the attention of the students in the next row back. “Take a hot marble and a cold marble, put ‘em together, pretty soon they’re the same temperature,” he said. Everyone nodded.
“But look at what happens in the universe. A star explodes and creates new elements. Gases cool and form solar systems and shit A planet cools, life emerges and crap, then evolves into fuckers like us. In thermodynamics, you physics fuckers act like each step cooler is a step towards annihilation, because we’re just a little bit closer to absolute zero. But each step was more highly ordered. Stars. Elements. Life. Not much order a second after the big bang. Just a bunch of hot quarks and shit. Now we have sex and basketball and mushrooms. Far more organized.” Everyone listening applauded. Stoney stood and curtseyed as though wearing a short skirt.
“Hi, everybody,” I said. “This is my friend Thomas Jackson. He’s auditing this class today. He’s smart.”
“I’m not as much smart as I am totally stoned,” said Stoney. At this point the door right of the lectern swung open suddenly and Professor Dannhausen strode in with a flourish.
“Rob, look. Just for today, can you sit just one seat over? I know you like being closer to Toni, but I promise you she’ll still love you if you let Stoney sit next to me today.”
“I never said I loved Rob,” said Toni, from my right.
“I promise you she won’t have sex with you any less if you just sit down,” I said. Rob sat down in the seat next to Stoney without comment.
“Don’t get pissed at him about this, Toni. This is my deal, and he’s doing me a favor.” We had, by this point, attracted Dr. Dannhausen’s attention.
“Mr. Baida, is class ready to begin?” Dr. Dannhausen asked me. He didn’t overstress anything. He wasn’t an asshole, he was just accustomed to starting his class on time.
“Yes, sir. Sorry about the delay.”
“And you have a new and colorfully attired student next to you?” said Dannhausen.
“Yes. This is my friend Thomas Jackson.”
“Yo. Call me Stoney, prof,” he said. Dannhausen cocked an eyebrow at me.
“He’s a Math major. It seems to me like there’s a lot of overlap between the two disciplines, and I thought he might be interested. He promises not to cause trouble,” I said.
“Absolutely. I’m a laid back, cool kind of mathematician,” said Stoney. He still hadn’t taken off his sunglasses.
“Okay. Welcome, Mr. Jackson. So, Mr. Baida, how about you explain to me and to your friend Mr. Jackson what Newton’s third law is.”
“Fuck. I totally know this,” said Stoney.
“Hush, Stoney,” I said, He put his hand to his mouth in embarrassment. “For every action there is an equal and opposite reaction,” I said.
“Right. Give me an example.”
“When I push on a brick, the brick pushes back.”
“Right. So let’s do a problem. Assume I have two canoes on a river, one with two occupants with a combined mass of 150 kilograms, and the other with three occupants with a combined mass of 250 kilos. The occupants push away from each other with a combined force of 43 newtons.”
“Weaklings,” said Stoney,” under his breath.
“How do we find the acceleration of the two canoes?” Prof. Dannhausen asked me. He didn’t usually focus on one student like this. He usually stated off by describing a principle, then explaining the basic concepts of the calculations, and then gave us a few calculations. He was zooming straight through to the end. In his defense, Newton’s third law is pretty basic.
“Well, according to the text, the acceleration is described as Σ Fx = max for ax.”
“According to the text?”
“Yes, sir,” I said.
“Do you not believe the text?” he asked.
“I’m not sure. The text never comes out and says these formulas are approximations, but it never comes out and says somebody’s sat down and tested them a thousand times and they were 100% accurate, either. I’m just not sure the math is more than a description of what we see.” All of the sudden I was back in Mr. Finch’s Sunday school class and I was challenging orthodoxy and realized I was about to get slapped. The other kids were pulling back from me, and I braced for it.
“Interesting, and possibly right,” said Prof. Dannhausen. “But let’s not get sidetracked. We’re talking about Newton’s second law and five people in two canoes. Let’s work through the problem.” No slap. Weird.
We talked through the problem. If you’re curious the second canoe is (and we’re solving the second one first because it has the greater mass):
ax = Σ F2x = 46 N = 0.18 m/s2
m2 250 kg
and the first canoe is
ax = Σ F1x = -46 N = -0.31 m/s2
m1 150 kg
and the only reason this problem exemplifies Newton’s third law is that the force, 46 Newtons, is applied to the formulae for both canoes—positive 46 newtons in the top equations (look for 46N in the numerator) and negative 46 newtons in the bottom. It took us a while to work through it. So you’ll notice that the same force applied to the lighter canoe caused it to accelerate more than the heavier canoe, because the heavier canoe is going in one direction at 0.18 meters per second, and the lighter one is going in the other direction at not quite twice as fast, or 0.31 meters per second. It took most of the class to work through all of the calculations and answer questions from the class, but at the end, the canoes were 0.35 meters apart.
“So is this a realistic calculation?” he asked the class.
“Oh, fuck, no,” said Stoney.
“No,” said Toni, to my right, loudly.
“Yes, Ms Sayers?”
“The solution ignored friction.”
“Exactly,” said prof Dannhausen.
“Lord fuck a duck,” said Stoney, in what he might have thought was a quiet voice. Prof. Dannhausen cocked an eyebrow at him.
“You think something else was left out of the solution, Mr. Jackson,” Prof. Dannhausen asked him.
“Well, yeah,” said Stoney. “You said the canoes were in a river. So there’s going to be current, and there’s no way two different canoes of different masses would react the same to it. And canoes are really unstable, and they wouldn’t react to being pushed like that. They'd spin around or something. But it would be really hard to get them to move in a straight line, so you'd have no way to calculate how far apart they'd be. This whole point mass concentration thing is a fiction. Kind of. I mean, I know I’m a guest and all, and I’ve enjoyed watching Henry and you solve the canoe deal, but it doesn’t really describe what would happen, does it?” Prof. Dannhausen put down his chalk and made a face that communicated “I wasn’t expecting this, but okay.” He thought for a minute before answering.
“Our visitor is largely correct. If you understood, please raise your hand.” Rob, Stoney, I, and Toni raised our hands, as did about a quarter of the class.
“First,” the professor said, “Most of the problems you will encounter this year assume that all of the masses involved are concentrated into points. This is assuredly not so for a canoe with two or three people in it. The center of gravity would be difficult to calculate, people move, changing things, if they spin, the centrifugal and centripetal forces would change. All of this would be …”
“Turbulent,” said Toni.
“Chaotic,” said Stoney.
“Both good descriptors. What I’m teaching you here is the basics of Newtonian dynamics and mechanics. Stripped to its bare bones. So the way Mr. Baida and I worked through the canoe problem was not realistic, and as you study more you will learn that mechanics is far more complicated than we are teaching here. But you need to understand the basics before we can develop the complexities. Forgive me if our problems are oversimplifications.” He paused and looked at the floor. “I also want to address something Mr. Baida said earlier. I believe that the equations and formulas you’re learning are accurate to a very precise degree. But no one can tell you that they’re true, because the fact that they describe reality accurately still leaves room for the notion that there may be another, different formula out there that describes things more accurately that we haven't yet discovered. It’s happened many times in physics that we thought we knew what we were looking at, then somebody came along and showed us we were wrong. But I’m not teaching you anything I know to be untrue.” He looked at his watch. “I think that’s all for today,” he said. “Mr. Jackson, come back any time.”
“Yeah, thanks,” he waved, and we all started standing up. “And none of you guys get stoned for this shit?” asked Stoney, to anyone in earshot.
“Whoah,” he said. “Fancy meeting you here.”
“I’m surprised to see you, too. For some reason I thought you weren’t an early riser.”
“Yea, well, actually, I’m not. For me this is more of a late night than an early morning. What are you up so early for?”
“Stoney, it’s 10:00.” I said.
“Yes?”
“Not the crack of dawn,” I said. He thought for a few seconds.
“So I take it you’re functional at this time of day with some frequency?” he asked.
“On my way to Physics. Happens three times a week.”
“Far out. Have fun, Physics. Man,” he said. He seemed happy for me.
“You want to come?”
“To a class?” he asked, dumbfounded.
“Sure. Physics. You might find it interesting. Plus, you’d almost certainly meet Rob and Toni.” He thought for a few seconds, as though confused.
“And this would interest me in some way?” he asked.
“They’re both completely insane, but smart. She’s gorgeous,” I said.
“Man, I don’t go to classes even when I’m registered for them. Suggesting I might go to one I’m not signed up for seems pretty fucked up.”
“Are you sober?” I asked. He thought about his answer before answering.
“’Sober’ would be inaccurate. In any sense of the word. But ‘sober’ generally refers to one’s interaction specifically with alcohol, and alcohol is not the largest component of my current weltanschauung.”
“How can you be fucked up at 10:00 a.m.?” I asked.
“Um, like I said, this is more of a late night than an early morning.”
“Did you take Physics at Lawrenceville?”
“Sure. It was a little silly,” said Stoney.
“How so?”
“All these extremely precise calculations that don’t relate to much reality.”
“How so?” I asked.
“Math doesn’t describe reality. Reality is, like reality. You get a bunch of guys standing around looking at cannon balls dropping from towers for a couple of hundred years, and eventually one of them invents a clock, then another one figures out how to put some numbers on a page that predict when it’s going to hit the ground. Later somebody calls it acceleration and gets famous for figuring out things about it. Then a hundred years later some other guy comes along and figures out a way to do the calculation that’s a little more accurate and he gets famous for that. It’s fucked up.” He paused and fumbled to find a cigarette, eventually found a soft pack of Marlboros somewhere in his boot, extracted one, and lit it with a disposable lighter. He looked up at me after taking the kid of puff a person who doesn’t smoke all the time takes.
“What were you saying?” he asked.
“You were explaining how Physics is messed up because people keep coming up with more precise formulae.”
“Oh, fuck yes. And every now and then somebody comes along and has a completely different way of looking at the problem of “what is acceleration?” and you guys all look at things differently. And then everybody in all the schools has to teach the new deal and everybody coalesces around Einstein. Or Bohr. Or who the fuck cares. You should read Kuhn. What?” I was looking at him and waiting for him to finish, and he was reacting to my expression.
“Nothing. I just get this a lot,” I said.
“What?”
“Math people telling me that physics is imprecise.”
“It fucking is imprecise. Has to be,” he said.
“And nothing like this happens in Math?” I asked.
“Oh, fuck no. Nobody ever came along and proved that 2+2=5. Nobody ever will. Nobody ever, since time began, came along and showed all the other mathematicians that the times tables were wrong, or that addition isn’t associative. Math just isn’t like that. Once something is so, it’s so for all time. Math just is.”
“I gotta go. Wanna come to class with me?” I asked.
“What’s class about?”
“Kinematics. Dynamics. Mechanics.” He thought a few seconds.
“I want to give you shit for talking like a physics guy, but I took high school physics so I know what all that means”
“So?”
“I say we blow it off and smoke some pot instead,” he said.
“Don’t smoke,” I said.
“Oh, right.” He thought for a few seconds. “Okay I’m game,” he said.
“Cool,” I said. “Your presence is going to surprise Toni and Rob.”
“Who are they?”
“They’re crazy people who sit on either side of me.”
“If it’s assigned seating, where are you going to put me?” he asked.
“No, it’s not assigned seating,” I said.
“So why do Rob and Toni always sit next to you?”
“They’re fucked up.”
“Like, stoned?” he asked.
“No. Like utterly insane,” I answered.
“I’m not completely stoned,” he said. “I should be able to deal. She’s the good-looking one, right?”
“Yes, but wacko like you don’t often encounter.”
“I’m not sure a person who doesn’t do drugs is qualified to make a statement like that,” he said. We were walking, more slowly than I usually do, towards the circular lecture hall where Physics met.
“Okay,” I answered.
“Like, once, when I was on acid, I was sure a red dragon with yellow dorsal plates and ivory teeth was licking the inside of my nose.”
“Ick,” I said, quoting Mary.
“It actually felt pretty good,” said Stoney, reflectively. “She, or maybe he, for that matter, what do I know about dragons? had a very soft and narrow tongue. Warm. It moved around in ... interesting ways.”
“Okay.” There was a brief reverie in which Stoney seemed at a loss for words. “Did you reciprocate?” I asked. He’d seemed pretty into it.
“How so?” he asked.
“Did you lick the red dragon’s nose?”
“Are you fucking nuts? Have you ever seen a dragon’s nose?” he asked.
“Um, no,” I answered, patiently, pretty sure that Stoney hadn’t seen one, either.
“Their noses are just like pigs’. Runny and dirty and noisy. No way I’m licking a dragon’s nose. That’s just gross.”
We had reached the amphitheater at the Science and Math Center. We were slightly, but not very, early. I took a seat in the middle of the hall, and Stony sat next to me.
“So why does Physics have to be taught in such a big room? A kind of a steep room.”
“It doesn’t, but this is a course all physics majors have to take, so it’s a big class, and this is a big classroom.”
“The physics guys and gals are actually pretty cute, now that I look at them,” he said, sitting down.
Right as he sat, Rob showed up on my left and Toni showed up on Stoney’s right.
“Excuse me, you’re in my seat,” Toni said to Stoney. Stoney, still in his sunglasses, looked up, confused.
“Excuse me?” he asked.
“You’re in my seat,” she said.
“Oh, right,” said Stoney. “You’re the one Henry told me about.”
“Look, Stoney, she’s eccentric. It will all go a lot easier if you sit on my left instead of my right.”
“What did he tell you about me?” Toni asked, a little more intensely than comfort allowed.
“Well,” Stoney said, noncommittally, looking at me.
“Feel free to be honest,” I said.
“He said you were gorgeous but … a little … eccentric.”
“Henry?” she asked.
“I said you were beautiful and brilliant but crazy,” I said. “Stoney, move to this seat,” I said, gesturing with my left hand. “It will be a little easier to deal with Rob.”
“Is this more of your sexist patronizing bullshit?” she asked.
“Yes,” I answered. “Stoney, sit over here.” Stoney stood and moved past me. Toni immediately sat down with too much ado. Students around us were unabashedly watching and laughing. Toni and Rob and I were dependable entertainment, and Stoney was a new addition to the act.
As Stoney passed me to sit on the other side, he paused for a second. “Her nipples got hard while she was yelling at us,” he whispered. “That’s so cool.”
“I heard that,” she said. Stoney shrugged.
“Hi,” said Rob, sticking out his hand towards Stoney. “I’m Rob.” He’d been standing there watching us, silently and patiently Rob-like, while we dealt with his girlfriend.
Stoney said “Yo. Rob. I’m Stoney,” and stuck out his hand. Rob looked at it, confused, then grasped it for a second.
“Is there a problem?” Rob asked, still standing.
“No. Rob. This is my friend Thomas Jackson. He, like me, is a math major.”
“You were a Physics major last time we talked,” said Rob.
“I’m a double major. Thomas—” I started.
“You can call me Stoney,” he said.
“Thanks,” said Rob and shook his hand again.
“Thomas--” I started.
“Stoney,” both Rob and Stoney corrected me, in unison.
“Stoney is a friend of mine from the Math department, so I invited him to class just to see what college Physics is like. So I know you like being close to Toni, but if this once you could sit one seat further to the left, we could keep Stoney up to date on what’s going on. Rob looked at him carefully.
“Did you take high school physics?” Rob asked.
“Sure,” he answered.
“What’s Newton’s second law?”
“Acceleration is proportional to force and inversely proportional to mass,” Stoney answered.
“Okay,” said Rob, although he didn’t sit down. He was still looking for a reason not to go along with the new seating scheme, Stoney shrugged and sat down between us.
“Henry gets very good grades bit he didn’t know the answer to that question last class, so far as I could tell,” said Rob, still standing. “Prof. Dannhausen doesn’t generally state first principles,” Rob said.
“That is a much simpler way of expressing the last set of problems,” I said, to Rob. I had, of course, been aware of Newton's laws last class, but there's no way I was getting in an argument with Rob, regardless of the topic.
“I’m just glad you didn’t ask me about the second law of thermodynamics ,” Stoney said. “That fucker’s fucked.”
“What?” said Rob and Toni, simultaneously.
“Yeah, it just is,” he answered.
“Why?” they demanded, simultaneously. Stoney was put off by their keen attention.
“Okay, well, it’s true that if a high energy system comes in contact with a low energy system, energy will flow from the higher to the other. To the lower.” Stoney looked up. He now had the attention of the students in the next row back. “Take a hot marble and a cold marble, put ‘em together, pretty soon they’re the same temperature,” he said. Everyone nodded.
“But look at what happens in the universe. A star explodes and creates new elements. Gases cool and form solar systems and shit A planet cools, life emerges and crap, then evolves into fuckers like us. In thermodynamics, you physics fuckers act like each step cooler is a step towards annihilation, because we’re just a little bit closer to absolute zero. But each step was more highly ordered. Stars. Elements. Life. Not much order a second after the big bang. Just a bunch of hot quarks and shit. Now we have sex and basketball and mushrooms. Far more organized.” Everyone listening applauded. Stoney stood and curtseyed as though wearing a short skirt.
“Hi, everybody,” I said. “This is my friend Thomas Jackson. He’s auditing this class today. He’s smart.”
“I’m not as much smart as I am totally stoned,” said Stoney. At this point the door right of the lectern swung open suddenly and Professor Dannhausen strode in with a flourish.
“Rob, look. Just for today, can you sit just one seat over? I know you like being closer to Toni, but I promise you she’ll still love you if you let Stoney sit next to me today.”
“I never said I loved Rob,” said Toni, from my right.
“I promise you she won’t have sex with you any less if you just sit down,” I said. Rob sat down in the seat next to Stoney without comment.
“Don’t get pissed at him about this, Toni. This is my deal, and he’s doing me a favor.” We had, by this point, attracted Dr. Dannhausen’s attention.
“Mr. Baida, is class ready to begin?” Dr. Dannhausen asked me. He didn’t overstress anything. He wasn’t an asshole, he was just accustomed to starting his class on time.
“Yes, sir. Sorry about the delay.”
“And you have a new and colorfully attired student next to you?” said Dannhausen.
“Yes. This is my friend Thomas Jackson.”
“Yo. Call me Stoney, prof,” he said. Dannhausen cocked an eyebrow at me.
“He’s a Math major. It seems to me like there’s a lot of overlap between the two disciplines, and I thought he might be interested. He promises not to cause trouble,” I said.
“Absolutely. I’m a laid back, cool kind of mathematician,” said Stoney. He still hadn’t taken off his sunglasses.
“Okay. Welcome, Mr. Jackson. So, Mr. Baida, how about you explain to me and to your friend Mr. Jackson what Newton’s third law is.”
“Fuck. I totally know this,” said Stoney.
“Hush, Stoney,” I said, He put his hand to his mouth in embarrassment. “For every action there is an equal and opposite reaction,” I said.
“Right. Give me an example.”
“When I push on a brick, the brick pushes back.”
“Right. So let’s do a problem. Assume I have two canoes on a river, one with two occupants with a combined mass of 150 kilograms, and the other with three occupants with a combined mass of 250 kilos. The occupants push away from each other with a combined force of 43 newtons.”
“Weaklings,” said Stoney,” under his breath.
“How do we find the acceleration of the two canoes?” Prof. Dannhausen asked me. He didn’t usually focus on one student like this. He usually stated off by describing a principle, then explaining the basic concepts of the calculations, and then gave us a few calculations. He was zooming straight through to the end. In his defense, Newton’s third law is pretty basic.
“Well, according to the text, the acceleration is described as Σ Fx = max for ax.”
“According to the text?”
“Yes, sir,” I said.
“Do you not believe the text?” he asked.
“I’m not sure. The text never comes out and says these formulas are approximations, but it never comes out and says somebody’s sat down and tested them a thousand times and they were 100% accurate, either. I’m just not sure the math is more than a description of what we see.” All of the sudden I was back in Mr. Finch’s Sunday school class and I was challenging orthodoxy and realized I was about to get slapped. The other kids were pulling back from me, and I braced for it.
“Interesting, and possibly right,” said Prof. Dannhausen. “But let’s not get sidetracked. We’re talking about Newton’s second law and five people in two canoes. Let’s work through the problem.” No slap. Weird.
We talked through the problem. If you’re curious the second canoe is (and we’re solving the second one first because it has the greater mass):
ax = Σ F2x = 46 N = 0.18 m/s2
m2 250 kg
and the first canoe is
ax = Σ F1x = -46 N = -0.31 m/s2
m1 150 kg
and the only reason this problem exemplifies Newton’s third law is that the force, 46 Newtons, is applied to the formulae for both canoes—positive 46 newtons in the top equations (look for 46N in the numerator) and negative 46 newtons in the bottom. It took us a while to work through it. So you’ll notice that the same force applied to the lighter canoe caused it to accelerate more than the heavier canoe, because the heavier canoe is going in one direction at 0.18 meters per second, and the lighter one is going in the other direction at not quite twice as fast, or 0.31 meters per second. It took most of the class to work through all of the calculations and answer questions from the class, but at the end, the canoes were 0.35 meters apart.
“So is this a realistic calculation?” he asked the class.
“Oh, fuck, no,” said Stoney.
“No,” said Toni, to my right, loudly.
“Yes, Ms Sayers?”
“The solution ignored friction.”
“Exactly,” said prof Dannhausen.
“Lord fuck a duck,” said Stoney, in what he might have thought was a quiet voice. Prof. Dannhausen cocked an eyebrow at him.
“You think something else was left out of the solution, Mr. Jackson,” Prof. Dannhausen asked him.
“Well, yeah,” said Stoney. “You said the canoes were in a river. So there’s going to be current, and there’s no way two different canoes of different masses would react the same to it. And canoes are really unstable, and they wouldn’t react to being pushed like that. They'd spin around or something. But it would be really hard to get them to move in a straight line, so you'd have no way to calculate how far apart they'd be. This whole point mass concentration thing is a fiction. Kind of. I mean, I know I’m a guest and all, and I’ve enjoyed watching Henry and you solve the canoe deal, but it doesn’t really describe what would happen, does it?” Prof. Dannhausen put down his chalk and made a face that communicated “I wasn’t expecting this, but okay.” He thought for a minute before answering.
“Our visitor is largely correct. If you understood, please raise your hand.” Rob, Stoney, I, and Toni raised our hands, as did about a quarter of the class.
“First,” the professor said, “Most of the problems you will encounter this year assume that all of the masses involved are concentrated into points. This is assuredly not so for a canoe with two or three people in it. The center of gravity would be difficult to calculate, people move, changing things, if they spin, the centrifugal and centripetal forces would change. All of this would be …”
“Turbulent,” said Toni.
“Chaotic,” said Stoney.
“Both good descriptors. What I’m teaching you here is the basics of Newtonian dynamics and mechanics. Stripped to its bare bones. So the way Mr. Baida and I worked through the canoe problem was not realistic, and as you study more you will learn that mechanics is far more complicated than we are teaching here. But you need to understand the basics before we can develop the complexities. Forgive me if our problems are oversimplifications.” He paused and looked at the floor. “I also want to address something Mr. Baida said earlier. I believe that the equations and formulas you’re learning are accurate to a very precise degree. But no one can tell you that they’re true, because the fact that they describe reality accurately still leaves room for the notion that there may be another, different formula out there that describes things more accurately that we haven't yet discovered. It’s happened many times in physics that we thought we knew what we were looking at, then somebody came along and showed us we were wrong. But I’m not teaching you anything I know to be untrue.” He looked at his watch. “I think that’s all for today,” he said. “Mr. Jackson, come back any time.”
“Yeah, thanks,” he waved, and we all started standing up. “And none of you guys get stoned for this shit?” asked Stoney, to anyone in earshot.
Tuesday, May 11, 2010
Chapter 19: A Number of Stumbling Blocks That Would Have Been Less So Had I Read Kuhn
After dinner I called Mrs. Wertheimer. Ginny had said I needed to call her, and she was right.
This was the first time I’d considered the idea that our relationship might have changed. When I met her, she’d been my teacher. Then, I’d bumped into her and she’d been nice enough to be my banker. At the time, I was an itinerant gambler and she’d facilitated my sinful and peripatetic ways. Now, I was a student again, I had identified her as my emergency contact, and I wasn’t entirely sure certain what the rules of our relationship were.
But even getting in touch with her wasn’t as easy as I anticipated. I dialed her number, using one plus dialing, and an operator came on the line.
“May I help you?” the operator asked.
“I’m just trying to call a friend in Chattanooga,” I said.
“I need your S.T.A.N. number,” the operator said.
“My what?”
“Your S.T.A.N. number,” she said.
“What’s that?”
“A Student Telephone Account Number,” she said.
“Don’t think I have one.”
“You’re a college student, right?” she asked.
“Yes,”
“Calling from (615) 555-1972?”
“Yes.”
“Before the semester began, your parents were sent a packet of information that allowed them to associate your long distance charges with another phone number as long as that phone number was a listed Bell System number,” she said, explaining the obvious.
“I don’t have any parents,” I said. An exaggeration, certainly, but I was tired of talking about it.
“Let’s see,” she said. “I show (615) 555-1972 as being assigned to Henry Baida.”
“That’s me.”
“And according to our records your mother is Margaret Wertheimer of Chattanooga, Tennessee,” she said.
“Interesting.”
“She’s not your mother?” she asked.
“No, not at all.”
“Well then who is she?” the operator asked, exasperated.
“Now, why is it you assume I’ll know who she is just because you don’t?” I asked. I was just curious.
“Sir, she’s associated with your phone number.”
“And your company’s … associations are never wrong?” I asked.
“They are reliable. Who is she?”
“Can I ask your name?” I asked.
“Nora. Operator 340F90D,” she said. “Who is Margaret Wertheimer?”
“She was my high school geometry teacher,” I said.
“And will she be willing to have your long distance charges associated with her phone number?”
“Meaning you’re going to bill her for my long distance charges?” I asked.
“Yes, of course.”
“Well, I guess she and I can work it out,” I said.
“Thank you. Connecting your call,” she said.
“Wait. What?”
“Connecting your call, sir,” she said, tired of me.
“You don’t need to confirm with her first that I’m going to be charging long distance calls to her number? I mean, I could have picked anybody.”
“I just don’t understand why you’re being so difficult,” said Nora, and I heard the phone start ringing, indicating that the call had gone through. I wondered if Nora planned to listen in. After a few rings, Mrs. W picked up.
“Hello?”
“Hello, Mrs. Wertheimer, this is Henry.”
Hello, Henry!” her voice brightened. “How in the world are you? How’s college?”
“Interesting. I’m learning stuff.”
“Like what? Tell me.”
“Physics, Greek, History, English.”
“And?”
“Physics is really cool. It’s easy, but I just haven’t thought about the universe that way before. All these precise mathematical formulas for working out the physical world,” I said. There was a slight pause.
“Precise?” she asked. I paused.
“Well, they’re teaching that the problems have exact answers.”
“What kind of problems?” she asked.
“Vectors, instantaneous acceleration, that kind of thing. Straight line stuff.” I could hear the scrape of her Zippo against the flint in the lighter and her first big drag.
“Look, Henry, I know it’s all fun. I remember my first physics class. But remember, none of the math is perfect. Newton came close, and people since have improved things. But when we think math, we’re people. When the universe acts itself out, it’s reality. You can’t describe the moment of a pendulum accurately with an equation any easier than you can do it with a sentence. You can say in a sentence that it swings back and forth. Equations aren’t much better, in some ways. They’re good, they’re just not perfect.” I paused to think about this.
“You’re saying the equations in my physics book don’t accurately describe pendulums?”
“Yeah. Sure. They’re approximations of reality. An equation explains the movement of a pendulum in the same way a paragraph does. But oscillators like pendulums are notoriously difficult to describe with precision.”
“I thought physics, the equations we’re studying, had been proven to accurately describe the physical universe.”
“Not really. All we can say is that they seem to accurately describe reality. But there’s no proof, the way there is in geometry. I can prove to you that opposite angles are equal. You can prove that the sum or product of any two even numbers will also be even. There’s nothing like that that in Physics. All you can do is say it seems to be accurate—what we predict by our equations seems to be pretty close to what happens when we measure it. No way to prove it.” I thought about that for a few seconds.
“Well, but if the equations always lead to accurate conclusions, surely that counts as proof.”
“Oh, good Lord no. Accuracy of prediction is meaningless in all ways,” she said. I could hear her pull on her cigarette.”
“That can’t be right,” I said.
“Henry, you can derive extraordinarily accurate predictions of the times and locations of solar and lunar eclipses if you assume that the Sun revolves around the earth. One reason the Ptolemaic system was so widely believed was the fact that it was extremely accurate in predicting events. Eclipses, transits, retrogrades—Ptolemy could predict them all. Of course his physical model, his explanation for why it was that the objects in the sky behaved as they did, didn’t match what we’ve learned since.”
“How did he explain retrograde motions of planets?” I asked.
“Planets had epicycles. Each one had a large orbit, more or less like what planetary orbits are like, only they were orbiting around the Earth. As was the Sun. But planets had a secondary sort of orbit, what was called an epicycle. The planets were kind of doing this curlicue in addition to their orbital motion.”
“Hard to imagine.”
“Think of the path of the Moon around the Earth, orbiting around as the Earth orbits the Sun.”
“Okay,” I said.
“Now imagine that exact path, the path the moon is taking, only the Earth isn’t there at the center of its orbit.”
“But the Moon wouldn’t trace that pattern if the Earth’s gravity didn’t anchor it.”
“Nobody knew anything about gravity back then. They thought that the planets and stars were attached to or moving in spheres,” she said.
“Scientists really believed in that kind of stuff? That wasn’t just popular mythology?
“Henry the only definition I’ve ever heard of ‘mythology’ that makes any sense to me is ‘mythology is other people’s religion.’ Yes, it was popular mythology, but Ptolemy lived in the second century, and educated people and astrologers believed in it too.”
“You mean astronomers?” I asked.
“Not really. There was no distinction in the second century. Ptolemy’s most famous book was about astrology. The fact that he could predict eclipses so accurately made his astrological forecasts especially prized.”
“This is weird,” I said.
“Why?” she seemed surprised that I was having a hard time with this.
“That a terracentric model could yield predictions similar to a heliocentric model. I would have thought, given orbital mechanics, which I admit I don’t understand at all, would have led the Ptolemaic idea to yield results that diverge from reality.”
There was a pause. I could tell from the sound of her exhaling that she was taking the last drag off of a cigarette and stubbing it out in an ashtray.
“Don’t start sounding too much like your professors, Henry,” she said.
“Yes ma’am.”
“Have you ever read an article by a college professor?”
“No, ma’am.”
“You knew every word you ever needed to use, except for words like ‘vector’ and ‘acceleration,’ the day Coach Pfieffer handed you your City High diploma,” she said.
“Yes ma’am.”
“So what were you saying?” she asked.
“I guess it seems odd that imagining that the Earth was the center of the Universe allowed Ptolemy to predict things so well.”
“It goes back way further than that. People before Ptolemy thought the Earth was flat and gods were driving chariots through the sky and that sort of thing. They could all still predict eclipses. Aristotle may have thought the world was round, I can’t remember, but he introduced the idea of the spheres in the heavens, and Ptolemy was considered reliable because his idea fit so neatly along with Aristotle’s, although astronomers of the day thought Ptolemy was a sort of amplification of Aristotle. Ptolemy made Aristotle more complex, but explained things that Aristotle didn’t. Planets moving in retrograde didn’t make sense with Aristotle’s spheres. But what I was going to say is that flat-earth people centuries before Christ were predicting celestial events like eclipses and planetary motion. All it takes to make predictions is persistence. Eventually you notice periodicity. Just because your predictions are accurate doesn’t men you understand what’s going on. Archimedes knew what time the sun was coming up.”
“Gack.”
“What, Henry?”
“Well, I’ve been operating on the supposition that what I was learning from teachers was objectively true. But it sounds like in Physics, anyway, what I’m learning is an approximation that has yet to be disproved. Did I understand?”
“Yes. That’s why I taught Math and not science.”
“And no matter how much I can trust the predictions of my calculations, what I’m taught may have foundations built on sand. Ptolemy looks good because he pretty much agrees with Aristotle, but then Copernicus comes along and blows them both out of the water.” There was a pause.
“Close. But when Copernicus came along nobody believed him, for almost a century,” she said.
“Why?”
“Because the old system worked so well. It predicted everything to fine detail. It explained their world view. It agreed with Christianity and Islam. All the pieces fit. When Copernicus published De revolutionibus orbium coelestium the academic and astronomical/astrological world pretty much ignored it.”
“He was an astrologer, too?”
“I don’t really know. That’s not clear. There’s no evidence he studied it. But he was a doctor, so most scholars think—“
“Copernicus was a doctor?”
“Yeah, sure. And a lawyer, too. And a priest. He spent a lot of time in college. Of course there wasn’t as much medicine as there is now. Or law, for that matter. But doctors were all expected to understand astrology.”
“Oh, for heaven’s sake.”
“Henry, this was the Fifteenth Century. Four hundred years ago. Four hundred years from now, everything we think is going to look just as silly.”
“Couldn’t possibly,” I said.
“Will certainly,” she answered.
“Gack.”
“Is it so terrible? Things change over time. Our understanding of the universe evolves, but will never be perfect.”
“No,” I answered. “Not terrible, but I just hadn’t thought about science this way, before. And this may give me some insight into something that sent me to college in the first place.”
“I thought you decided to go to college because you were loaded in Taccoa.”
“Yeah, but what had been worrying me is this variation in the way the universe works. I perceive it as a breakdown in the rules.”
“You’re not loaded now, are you?”
“No, no. You should know me better than that. What gets me is that you do the same thing a thousand times in exactly the same way, and three of them yield different results.”
“Okay,” she said, taking a drag.
“I have this working theory that the rules of the universe have imperfections, but those imperfections don’t lead to intolerably divergent results, so most people ignore the exceptions, the three out of a thousand. Odd to think that the rules of the universe have flaws, but odder still that most people don’t notice. When something weird happens, witnesses chalk it up to human factors and variables that are impossible to control. People expect strangeness and unpredictability, so shake it off when they see it.”
“And that’s not how you see the cosmos?” she asked. “Imperfect rules with unperceived flaws?” I thought for a few seconds.
“To me it’s always seemed as though the rules of the universe were an enormous checkerboard, stretching off into infinity in all directions. It’s regular, all perfect squares, and the sides of the squares are less than a millimeter per side. That’s the Universe as it should be. A plane of tiny squares stretching off in all directions.”
“Okay.” I could hear not only another cigarette being lit but the clink of a bottle on the rim of a glass.
“Mrs. W, are you having a drink?”
“Yes, of course,” she said. “B and B. I’m not a savage. Go on with your checkerboard.”
“Okay. If you mark off a grid, several hundred squares, and then just start blackening them, checkerboard style, working outward from three or four randomly selected squares, most of the time, they’ll all link up into an orderly checkerboard. They’ll all agree and your checkerboard will stretch out to infinity. Sometimes, though, one of them will be out of sync with the others, and you’ll get this odd pattern of darkened squares.”
“How do you know this?”
“My math class is really boring. I doodle a lot.”
“Go on,” she said.
“So what happens is that the squares that were out of sync with the rest of the grid get sealed off. There’s a little pattern of out of sync squares surrounded by like a dark circle, a sort of city wall, like cities had in the days before cannons. But the pattern prevails. There’s a checkerboard isolated within a checkerboard. Sometimes you get these fault line-like things, but the overall checkerboard pattern always wins out.”
“Okay.”
“Can you imagine my two-dimensional checkerboard?”
“I think so.”
“Okay now think of the irregularities and the checkerboard patterns as in three dimensions.” She took a drag of her cigarette.
“Okay, I got it. The irregularities are encysted.”
“Exactly. So that three-D checkerboard is a representation of life in the same way you taught me to draw a three-dimensional graph on a two-dimensional piece of paper. Inaccurate, missing a dimension, but revealing.”
“Interesting,” she said. “Metaphysical, and therefore dubious, but interesting. So you think something’s going wrong with the universe, the rules don’t always work perfectly, but that these errors either seal themselves off or are sealed off by the fact that the rest of the universe is working according to the rules. It’s all on grid?”
“You got it,” I said. It felt good to be understood.
“It’s interesting to think about, but you’re talking about a way of organizing the universe that nobody’s going to see any evidence for.”
“True enough.”
“Then tell me something else,” she said. “Made any friends?”
“Some. There are a few guys on my floor, in my dorm, that seem like good guys. I’m a little older than the other freshmen, but some of them seem okay. And I ran into Ginny earlier today, and she reminded me to call you.”
“I was wondering if you’d run into her. Peabody’s right across the street.”
“Why’d you recommend Peabody for her and not for me?”
“Most of her family went there. Her mother went there. We came from farm people in Warren County, Tennessee. They went to get a good rural education when college was still relatively cheap.”
“What’s a rural education?” I asked.
“Peabody used to have the world’s most productive dairy heard. They taught people how to be the best farmers in the U.S. Ginny’s grandparents went there and learned that.”
“Odd. Mainly what she talked about to me was sports.”
“She loves tennis. But that dairy herd deal runs deep in her family. Until Gunner,” she said.
“Gunner?” I asked.
“Her father. A lawyer. Can’t make up my mind about him.” I was about to ask about Gunner, but she turned conversation back to me. “So what kinds of friends?”
“On my dorm floor? Well, there’s Milton, from California, and Joel Bernstein from White Plains, New York. Brian Wilmot from Cherry Hill, New Jersey. Our R.A. is a psychology grad student from someplace in Louisiana I can’t pronounce.”
“A good mix?” she asked.
“Yes, ma’am. There are lots more Southerners than people from elsewhere, but there are people here from all over. Mostly white, upper middle class. Some I would call rich. Michael Scott is black, and José Sanchez is Cuban. They say there’s a guy on our floor somewhere who’s a Viet Nam vet, but I haven’t met him.”
“How are your courses?”
“Greek is wonderful. Physics is interesting, but they’re going really, really slowly. Math is boring. History could be interesting, but the prof is letting his politics affect his analysis.”
“That’s stupid,” she said.
“He would say that history has always had, seen or unseen, known or unknown, a bias towards capitalism, and the fact that he admits to the influence of Marxism on his thinking and presentation merely makes him a more honest purveyor of American History than has traditionally been the case.”
“One of those.”
“Very much one of those. I think professors should be like journalists, and strain against their personal views, rather than giving in to them. At least professionally.”
“And judges,” she said, lighting another Benson & Hedges.
“Hmmm,” I mused.
“You don’t think judges should strain against personal prejudice?” she asked.
“No, ma’am, I do. It’s just in my experience, they don’t.”
“Ed Morgan did paste you one, that’s for sure,” she said. “But, really, Henry, calling yourself Leon Trotsky. Why in the world?”
“Oh, it happened in a P.G. Wodehouse story, and so it was the first thing that popped into my head.”
“You meeting any girls?”
“Well, there’s Toni from Physics. Beautiful but barking mad. In an exceedingly strange relationship with Rob. Both physics majors. Then there’s Mary Roberts from Greek. She’s very serious about her religion, seems to be Tri-Delt material , and seems taken with Brian.”
“From Cherry Hill?”
“Yes,” I said, surprised she remembered.
“Does he have money?”
“Seems to.”
“A lot of money?” she asked.
“Maybe. There’s a family sportswear business. She likes his uniform, too, I gather.”
“What kind?”
“N.R.O.T.C.”
“Well, sailors do have the best uniforms. What’s wrong with math?” she asked.
“It’s just very basic.”
“What are you studying?”
“Today it was horizontal transformations.” I could hear her whistling out her cigarette smoke in disapproval.
“Who’s your professor?” she asked.
“Anton Ladd,” I answered.
“I know him,” she said. “Kind of a pain in the ass. Like maybe he’s really constipated. Very stuck on protocol. But he’s smart. Does good work.”
“How do you know him?”
“He was at some of the early conferences on turbulence theory. He was in the audience once when I delivered a paper on derivatives of binary sequences, then I was in the audience when he presented some number theory deal. Complimenting sets of n-tuples of integers, or something like that.”
“Mrs. W., you’re full of surprises. I had no idea you’d been presenting papers at mathematical conferences,” I said. I could almost hear her shrug.
“Everybody needs a hobby,” she said.
“Ladd kind of seemed to know who you were and almost wanted to doubt you’d been my teacher. Interesting to hear how he knows you. But speaking of number theory, there’s this guy in my math class. I almost never see him because he refuses to come to class except for tests.”
“Where’s he from?’
“I’m not sure. He went to school in someplace called Lawrenceville, New Jersey, but it was a boarding school, I think, so I’m not sure where he’s from. Had a really good math program in high school and can’t stand our professor. Calls himself Stoney on account of his name, he says.”
“How’s he doing?”
“Only one test so far. He and I had the only 100s. The prof is plainly irritated by this. Not by me. I come to class and all. But he really hates Stoney.”
“Congratulations on the 100, Henry.”
“This is stuff you had me doing junior year, but thanks. Anyway, so Stoney wants to start a math club.”
“To do what?”
“He knows these other two guys who are interested in math and he wants to start a group to take on something more challenging.” I could hear her smoking and thinking about this.
“What would you do?” My turn to shrug.
“Something hard,” I said.
“Like what?”
“The only thing he mentioned was to see if we could deduce Kepler’s laws from Tycho’s observations.”
“Oh, gack, no. Henry, don’t join.”
“Why?” Unlike her to discourage an interest in Math.
“They were both crazy as coots,” she said. “Tycho was this nobleman with a common-law wife and he had all these extremely precise measurements of planetary positions because he was rich and could spend huge amounts of money on assistants and astronomical instruments. But he had no math whatsoever. He was a nobleman. Practical skills were beneath him.”
“So?”
“Tycho’s theory was that the sun and the moon and the stars revolved around the earth, and that the five planets— ”
“Five?”
“Mercury, Venus, Mars, Jupiter, and Saturn were it, back then. Uranus and Neptune weren’t discovered until there were really good telescopes, and Tycho didn’t have a telescope. Pluto was deduced from wobbles in the orbits of other planets, and Earth wasn’t regarded as a planet back then”
“Why is Pluto a planet? It’s not like the others,” I said.
“It’s out there. You leave Pluto alone.”
“Okay, Then if Earth wasn’t a planet, what was it?”
“The center of the universe.”
“They hadn’t read Copernicus?” I asked.
“Sure, they read it, but nobody believed him. Copernicus disagreed with what they’d been taught, and what they’d been taught explained everything pretty well. Plus, Tycho was really, really egotistical. He wanted his theory to be the one that explained the universe. He thought the Earth was the center of the universe, that the sun revolved around the Earth, and all the other planets revolved around the sun.”
“That’s insane.”
“No it’s not. If the universe were constructed like that, it would look exactly like it does now, for all intents and purposes. Think it through, later on. Imagine the Earth is stationary, and everything else is the way Tycho thought. It’s more complicated to imagine, but it would look exactly the same. It may actually be that way—there’s no way to know. But, anyway, Tycho had all these amazingly precise observations, but he had no math. No way to analyze all that data. So he brings Kepler from Austria to Prague, thinks of him as a hireling, and won’t share his data with him. Not really. He shows him some of his Mars stuff, because he’s convinced they support his inane Earth-Sun deal. Then Tycho dies, and his heirs are worse even than Tycho was. It’s all a mess.”
“But didn’t Kepler deduce the laws of planetary motion from Tycho’s observations?” I asked.
“Sort of.”
“Sort of?”
“’Deduce’ isn’t the right word. ‘Stumbled onto’ is better. Once Kepler at long last got access to Tycho’s log books—Tycho’s dead now, remember—he made some really clever deductions. But the only reason he did was that he made a huge calculation error in the orbit of Mars. Because of that he went on and recalculated Mars’ orbital path a number of times, and eventually realized that it wasn’t circular. But if he hadn’t made a huge error in the first calculation, he wouldn’t have kept calculating. All of Kepler’s laws came from that mistake, so it’s a good mistake.”
“So why do you think we should avoid Kepler and Tycho as a project?” I asked.
“They were all nuts,” she said. “Kepler thought his mother was a witch. Tycho complained that the King of Denmark didn’t value his services enough, and complained that the emperor or whatever he was in Prague wasn’t providing him a grand enough castle. They both cast horoscopes and believed in them.”
“Crazy people sometimes have great ideas,” I said.
“More likely they don’t,” she answered. “Most crazy people have crazy ideas. But the reason for avoiding a club that wants to deduce Kepler from Tycho is that it would be extraordinarily boring. You wouldn’t recognize any of Kepler’s calculations as Math.”
“What were they?” I asked.
“Arithmetic,” she answered. “Liebnitz and Newton and the calculus you like so much came in the next century. Kepler just sat in his little apartment in Prague and re-did the calculations over and over and over until they came out right.”
“πr2 all the way?” I asked.
“Yes. Each guess took thousands of calculations. All multiplication and addition. No functions at all. A complete pain in the … neck. You don’t want to spend time on that kind of thing. Months and months of rote calculation.”
“Okay. So , assuming there’s a club, what should we do instead, then?”
“Maybe something with Maxwell.”
“Okay,” I said, hesitantly.
“You don’t like John Clerk Maxwell?” she asked.
“Wasn’t he a physicist?” I asked.
“Yes, of course.”
“Well, the rest of these guys are math guys. I may be the only physics guy.”
“Then maybe the Lorentz transformations,” she said.
“What’s that?”
“Hendrick Lorentz described a set of functions (math functions that cannot be priduced in blogspot here, if you want a copy, send me a note)
where
(a short but complex equation should be here; can't insert either objects or pictures as text),” she said. “As best I can recall.” I thought that through for a moment.
“Leaving y-prime and z-prime alone for now, that would move around a lot,” I said, after a few seconds.
“No kidding.”
“I’m having a hard time imagining the graph. It must fold back on itself a lot.”
“It moves. It changes from 45 degrees off the x axis to 45 degrees off the y axis and then back again.”
“Cool. Is it good for anything?”
“Lorentz came up with it as a way to explain the propagation of light through the luminous æther.”
“The what?”
“The luminous æther. Going back to Aristotle, everybody assumed there was something that filled space, in a way. You’ve heard of fire, air, water, and earth. Aristotle said there was a fifth element that gave definition to space. In the nineteenth century there were a lot of discoveries that indicated that light was a wave of some sort, so a lot of physicists decided there must be some sort of medium the waves were passing through.”
“Why does there have to be a medium?” I asked.
“Now you’re thinking like Maxwell,” she said.
“Why?”
“Never mind. Up until Heisenberg and Bohr and those guys, we always thought about Physics in terms of physical realities. That’s why it’s a called Physics.”
“Oddly enough, a beautiful but crazy girl said that to me recently,” I said.
“Oh, yeah? Are you interested in her?”
“No, no. To me the crazy comes through much more loudly than the beautiful,” I answered. “So why does there have to be a medium?”
“Well, they thought the closest analogy to the way light moves was the way sound moves through the air, and they though that was because of regular compressions moving through the air. The compressions graph out to be a wave, even though they’re something else. If light was a wave, there had to be something for it to be compressing. A wave seems to need something to be moving through. So they referred back to Aristotle and called the stuff æther.”
“All interesting. Fascinating, in fact,” I said. “You are ever the teacher. So why are Lorentz’ calculations important? I’m thinking people probably don’t still believe in the æther.”
“No, but people did into the Twenties. Albert saw the Lorentz transformations as an accurate description of a misunderstood process and deduced Special Relativity from them.”
“Albert Einstein?”
“Yes,” she said. I could her exhale smoke.”
“I’ll suggest it to the guys but I’m the only physics student in the proposed club. There seems to be this odd hostility between mathematics and physics.”
“Well, that comes and goes. Look, Henry, I’m sorry, but I need to get off the phone. Are you set for money?”
“Oh, sure.”
“Are you coming home for Thanksgiving?”
“Geez. I hadn’t thought about it. I don’t know where my parents or my sister are.”
“No, Henry, I meant are you coming to home Chattanooga? Do you want to have Thanksgiving with us?”
“Well, sure, thanks. I don’t have a car, though.”
“So?”
“I guess I can take a cab from the Green Ghetto to your house. And I can take the bus from Nashville to Chattanooga.”
“Henry, for Heaven’s sake. Don’t make it complicated. You can stay with me. Call Ginny and you two come down together. I’m sorry. I really have to run.”
This was the first time I’d considered the idea that our relationship might have changed. When I met her, she’d been my teacher. Then, I’d bumped into her and she’d been nice enough to be my banker. At the time, I was an itinerant gambler and she’d facilitated my sinful and peripatetic ways. Now, I was a student again, I had identified her as my emergency contact, and I wasn’t entirely sure certain what the rules of our relationship were.
But even getting in touch with her wasn’t as easy as I anticipated. I dialed her number, using one plus dialing, and an operator came on the line.
“May I help you?” the operator asked.
“I’m just trying to call a friend in Chattanooga,” I said.
“I need your S.T.A.N. number,” the operator said.
“My what?”
“Your S.T.A.N. number,” she said.
“What’s that?”
“A Student Telephone Account Number,” she said.
“Don’t think I have one.”
“You’re a college student, right?” she asked.
“Yes,”
“Calling from (615) 555-1972?”
“Yes.”
“Before the semester began, your parents were sent a packet of information that allowed them to associate your long distance charges with another phone number as long as that phone number was a listed Bell System number,” she said, explaining the obvious.
“I don’t have any parents,” I said. An exaggeration, certainly, but I was tired of talking about it.
“Let’s see,” she said. “I show (615) 555-1972 as being assigned to Henry Baida.”
“That’s me.”
“And according to our records your mother is Margaret Wertheimer of Chattanooga, Tennessee,” she said.
“Interesting.”
“She’s not your mother?” she asked.
“No, not at all.”
“Well then who is she?” the operator asked, exasperated.
“Now, why is it you assume I’ll know who she is just because you don’t?” I asked. I was just curious.
“Sir, she’s associated with your phone number.”
“And your company’s … associations are never wrong?” I asked.
“They are reliable. Who is she?”
“Can I ask your name?” I asked.
“Nora. Operator 340F90D,” she said. “Who is Margaret Wertheimer?”
“She was my high school geometry teacher,” I said.
“And will she be willing to have your long distance charges associated with her phone number?”
“Meaning you’re going to bill her for my long distance charges?” I asked.
“Yes, of course.”
“Well, I guess she and I can work it out,” I said.
“Thank you. Connecting your call,” she said.
“Wait. What?”
“Connecting your call, sir,” she said, tired of me.
“You don’t need to confirm with her first that I’m going to be charging long distance calls to her number? I mean, I could have picked anybody.”
“I just don’t understand why you’re being so difficult,” said Nora, and I heard the phone start ringing, indicating that the call had gone through. I wondered if Nora planned to listen in. After a few rings, Mrs. W picked up.
“Hello?”
“Hello, Mrs. Wertheimer, this is Henry.”
Hello, Henry!” her voice brightened. “How in the world are you? How’s college?”
“Interesting. I’m learning stuff.”
“Like what? Tell me.”
“Physics, Greek, History, English.”
“And?”
“Physics is really cool. It’s easy, but I just haven’t thought about the universe that way before. All these precise mathematical formulas for working out the physical world,” I said. There was a slight pause.
“Precise?” she asked. I paused.
“Well, they’re teaching that the problems have exact answers.”
“What kind of problems?” she asked.
“Vectors, instantaneous acceleration, that kind of thing. Straight line stuff.” I could hear the scrape of her Zippo against the flint in the lighter and her first big drag.
“Look, Henry, I know it’s all fun. I remember my first physics class. But remember, none of the math is perfect. Newton came close, and people since have improved things. But when we think math, we’re people. When the universe acts itself out, it’s reality. You can’t describe the moment of a pendulum accurately with an equation any easier than you can do it with a sentence. You can say in a sentence that it swings back and forth. Equations aren’t much better, in some ways. They’re good, they’re just not perfect.” I paused to think about this.
“You’re saying the equations in my physics book don’t accurately describe pendulums?”
“Yeah. Sure. They’re approximations of reality. An equation explains the movement of a pendulum in the same way a paragraph does. But oscillators like pendulums are notoriously difficult to describe with precision.”
“I thought physics, the equations we’re studying, had been proven to accurately describe the physical universe.”
“Not really. All we can say is that they seem to accurately describe reality. But there’s no proof, the way there is in geometry. I can prove to you that opposite angles are equal. You can prove that the sum or product of any two even numbers will also be even. There’s nothing like that that in Physics. All you can do is say it seems to be accurate—what we predict by our equations seems to be pretty close to what happens when we measure it. No way to prove it.” I thought about that for a few seconds.
“Well, but if the equations always lead to accurate conclusions, surely that counts as proof.”
“Oh, good Lord no. Accuracy of prediction is meaningless in all ways,” she said. I could hear her pull on her cigarette.”
“That can’t be right,” I said.
“Henry, you can derive extraordinarily accurate predictions of the times and locations of solar and lunar eclipses if you assume that the Sun revolves around the earth. One reason the Ptolemaic system was so widely believed was the fact that it was extremely accurate in predicting events. Eclipses, transits, retrogrades—Ptolemy could predict them all. Of course his physical model, his explanation for why it was that the objects in the sky behaved as they did, didn’t match what we’ve learned since.”
“How did he explain retrograde motions of planets?” I asked.
“Planets had epicycles. Each one had a large orbit, more or less like what planetary orbits are like, only they were orbiting around the Earth. As was the Sun. But planets had a secondary sort of orbit, what was called an epicycle. The planets were kind of doing this curlicue in addition to their orbital motion.”
“Hard to imagine.”
“Think of the path of the Moon around the Earth, orbiting around as the Earth orbits the Sun.”
“Okay,” I said.
“Now imagine that exact path, the path the moon is taking, only the Earth isn’t there at the center of its orbit.”
“But the Moon wouldn’t trace that pattern if the Earth’s gravity didn’t anchor it.”
“Nobody knew anything about gravity back then. They thought that the planets and stars were attached to or moving in spheres,” she said.
“Scientists really believed in that kind of stuff? That wasn’t just popular mythology?
“Henry the only definition I’ve ever heard of ‘mythology’ that makes any sense to me is ‘mythology is other people’s religion.’ Yes, it was popular mythology, but Ptolemy lived in the second century, and educated people and astrologers believed in it too.”
“You mean astronomers?” I asked.
“Not really. There was no distinction in the second century. Ptolemy’s most famous book was about astrology. The fact that he could predict eclipses so accurately made his astrological forecasts especially prized.”
“This is weird,” I said.
“Why?” she seemed surprised that I was having a hard time with this.
“That a terracentric model could yield predictions similar to a heliocentric model. I would have thought, given orbital mechanics, which I admit I don’t understand at all, would have led the Ptolemaic idea to yield results that diverge from reality.”
There was a pause. I could tell from the sound of her exhaling that she was taking the last drag off of a cigarette and stubbing it out in an ashtray.
“Don’t start sounding too much like your professors, Henry,” she said.
“Yes ma’am.”
“Have you ever read an article by a college professor?”
“No, ma’am.”
“You knew every word you ever needed to use, except for words like ‘vector’ and ‘acceleration,’ the day Coach Pfieffer handed you your City High diploma,” she said.
“Yes ma’am.”
“So what were you saying?” she asked.
“I guess it seems odd that imagining that the Earth was the center of the Universe allowed Ptolemy to predict things so well.”
“It goes back way further than that. People before Ptolemy thought the Earth was flat and gods were driving chariots through the sky and that sort of thing. They could all still predict eclipses. Aristotle may have thought the world was round, I can’t remember, but he introduced the idea of the spheres in the heavens, and Ptolemy was considered reliable because his idea fit so neatly along with Aristotle’s, although astronomers of the day thought Ptolemy was a sort of amplification of Aristotle. Ptolemy made Aristotle more complex, but explained things that Aristotle didn’t. Planets moving in retrograde didn’t make sense with Aristotle’s spheres. But what I was going to say is that flat-earth people centuries before Christ were predicting celestial events like eclipses and planetary motion. All it takes to make predictions is persistence. Eventually you notice periodicity. Just because your predictions are accurate doesn’t men you understand what’s going on. Archimedes knew what time the sun was coming up.”
“Gack.”
“What, Henry?”
“Well, I’ve been operating on the supposition that what I was learning from teachers was objectively true. But it sounds like in Physics, anyway, what I’m learning is an approximation that has yet to be disproved. Did I understand?”
“Yes. That’s why I taught Math and not science.”
“And no matter how much I can trust the predictions of my calculations, what I’m taught may have foundations built on sand. Ptolemy looks good because he pretty much agrees with Aristotle, but then Copernicus comes along and blows them both out of the water.” There was a pause.
“Close. But when Copernicus came along nobody believed him, for almost a century,” she said.
“Why?”
“Because the old system worked so well. It predicted everything to fine detail. It explained their world view. It agreed with Christianity and Islam. All the pieces fit. When Copernicus published De revolutionibus orbium coelestium the academic and astronomical/astrological world pretty much ignored it.”
“He was an astrologer, too?”
“I don’t really know. That’s not clear. There’s no evidence he studied it. But he was a doctor, so most scholars think—“
“Copernicus was a doctor?”
“Yeah, sure. And a lawyer, too. And a priest. He spent a lot of time in college. Of course there wasn’t as much medicine as there is now. Or law, for that matter. But doctors were all expected to understand astrology.”
“Oh, for heaven’s sake.”
“Henry, this was the Fifteenth Century. Four hundred years ago. Four hundred years from now, everything we think is going to look just as silly.”
“Couldn’t possibly,” I said.
“Will certainly,” she answered.
“Gack.”
“Is it so terrible? Things change over time. Our understanding of the universe evolves, but will never be perfect.”
“No,” I answered. “Not terrible, but I just hadn’t thought about science this way, before. And this may give me some insight into something that sent me to college in the first place.”
“I thought you decided to go to college because you were loaded in Taccoa.”
“Yeah, but what had been worrying me is this variation in the way the universe works. I perceive it as a breakdown in the rules.”
“You’re not loaded now, are you?”
“No, no. You should know me better than that. What gets me is that you do the same thing a thousand times in exactly the same way, and three of them yield different results.”
“Okay,” she said, taking a drag.
“I have this working theory that the rules of the universe have imperfections, but those imperfections don’t lead to intolerably divergent results, so most people ignore the exceptions, the three out of a thousand. Odd to think that the rules of the universe have flaws, but odder still that most people don’t notice. When something weird happens, witnesses chalk it up to human factors and variables that are impossible to control. People expect strangeness and unpredictability, so shake it off when they see it.”
“And that’s not how you see the cosmos?” she asked. “Imperfect rules with unperceived flaws?” I thought for a few seconds.
“To me it’s always seemed as though the rules of the universe were an enormous checkerboard, stretching off into infinity in all directions. It’s regular, all perfect squares, and the sides of the squares are less than a millimeter per side. That’s the Universe as it should be. A plane of tiny squares stretching off in all directions.”
“Okay.” I could hear not only another cigarette being lit but the clink of a bottle on the rim of a glass.
“Mrs. W, are you having a drink?”
“Yes, of course,” she said. “B and B. I’m not a savage. Go on with your checkerboard.”
“Okay. If you mark off a grid, several hundred squares, and then just start blackening them, checkerboard style, working outward from three or four randomly selected squares, most of the time, they’ll all link up into an orderly checkerboard. They’ll all agree and your checkerboard will stretch out to infinity. Sometimes, though, one of them will be out of sync with the others, and you’ll get this odd pattern of darkened squares.”
“How do you know this?”
“My math class is really boring. I doodle a lot.”
“Go on,” she said.
“So what happens is that the squares that were out of sync with the rest of the grid get sealed off. There’s a little pattern of out of sync squares surrounded by like a dark circle, a sort of city wall, like cities had in the days before cannons. But the pattern prevails. There’s a checkerboard isolated within a checkerboard. Sometimes you get these fault line-like things, but the overall checkerboard pattern always wins out.”
“Okay.”
“Can you imagine my two-dimensional checkerboard?”
“I think so.”
“Okay now think of the irregularities and the checkerboard patterns as in three dimensions.” She took a drag of her cigarette.
“Okay, I got it. The irregularities are encysted.”
“Exactly. So that three-D checkerboard is a representation of life in the same way you taught me to draw a three-dimensional graph on a two-dimensional piece of paper. Inaccurate, missing a dimension, but revealing.”
“Interesting,” she said. “Metaphysical, and therefore dubious, but interesting. So you think something’s going wrong with the universe, the rules don’t always work perfectly, but that these errors either seal themselves off or are sealed off by the fact that the rest of the universe is working according to the rules. It’s all on grid?”
“You got it,” I said. It felt good to be understood.
“It’s interesting to think about, but you’re talking about a way of organizing the universe that nobody’s going to see any evidence for.”
“True enough.”
“Then tell me something else,” she said. “Made any friends?”
“Some. There are a few guys on my floor, in my dorm, that seem like good guys. I’m a little older than the other freshmen, but some of them seem okay. And I ran into Ginny earlier today, and she reminded me to call you.”
“I was wondering if you’d run into her. Peabody’s right across the street.”
“Why’d you recommend Peabody for her and not for me?”
“Most of her family went there. Her mother went there. We came from farm people in Warren County, Tennessee. They went to get a good rural education when college was still relatively cheap.”
“What’s a rural education?” I asked.
“Peabody used to have the world’s most productive dairy heard. They taught people how to be the best farmers in the U.S. Ginny’s grandparents went there and learned that.”
“Odd. Mainly what she talked about to me was sports.”
“She loves tennis. But that dairy herd deal runs deep in her family. Until Gunner,” she said.
“Gunner?” I asked.
“Her father. A lawyer. Can’t make up my mind about him.” I was about to ask about Gunner, but she turned conversation back to me. “So what kinds of friends?”
“On my dorm floor? Well, there’s Milton, from California, and Joel Bernstein from White Plains, New York. Brian Wilmot from Cherry Hill, New Jersey. Our R.A. is a psychology grad student from someplace in Louisiana I can’t pronounce.”
“A good mix?” she asked.
“Yes, ma’am. There are lots more Southerners than people from elsewhere, but there are people here from all over. Mostly white, upper middle class. Some I would call rich. Michael Scott is black, and José Sanchez is Cuban. They say there’s a guy on our floor somewhere who’s a Viet Nam vet, but I haven’t met him.”
“How are your courses?”
“Greek is wonderful. Physics is interesting, but they’re going really, really slowly. Math is boring. History could be interesting, but the prof is letting his politics affect his analysis.”
“That’s stupid,” she said.
“He would say that history has always had, seen or unseen, known or unknown, a bias towards capitalism, and the fact that he admits to the influence of Marxism on his thinking and presentation merely makes him a more honest purveyor of American History than has traditionally been the case.”
“One of those.”
“Very much one of those. I think professors should be like journalists, and strain against their personal views, rather than giving in to them. At least professionally.”
“And judges,” she said, lighting another Benson & Hedges.
“Hmmm,” I mused.
“You don’t think judges should strain against personal prejudice?” she asked.
“No, ma’am, I do. It’s just in my experience, they don’t.”
“Ed Morgan did paste you one, that’s for sure,” she said. “But, really, Henry, calling yourself Leon Trotsky. Why in the world?”
“Oh, it happened in a P.G. Wodehouse story, and so it was the first thing that popped into my head.”
“You meeting any girls?”
“Well, there’s Toni from Physics. Beautiful but barking mad. In an exceedingly strange relationship with Rob. Both physics majors. Then there’s Mary Roberts from Greek. She’s very serious about her religion, seems to be Tri-Delt material , and seems taken with Brian.”
“From Cherry Hill?”
“Yes,” I said, surprised she remembered.
“Does he have money?”
“Seems to.”
“A lot of money?” she asked.
“Maybe. There’s a family sportswear business. She likes his uniform, too, I gather.”
“What kind?”
“N.R.O.T.C.”
“Well, sailors do have the best uniforms. What’s wrong with math?” she asked.
“It’s just very basic.”
“What are you studying?”
“Today it was horizontal transformations.” I could hear her whistling out her cigarette smoke in disapproval.
“Who’s your professor?” she asked.
“Anton Ladd,” I answered.
“I know him,” she said. “Kind of a pain in the ass. Like maybe he’s really constipated. Very stuck on protocol. But he’s smart. Does good work.”
“How do you know him?”
“He was at some of the early conferences on turbulence theory. He was in the audience once when I delivered a paper on derivatives of binary sequences, then I was in the audience when he presented some number theory deal. Complimenting sets of n-tuples of integers, or something like that.”
“Mrs. W., you’re full of surprises. I had no idea you’d been presenting papers at mathematical conferences,” I said. I could almost hear her shrug.
“Everybody needs a hobby,” she said.
“Ladd kind of seemed to know who you were and almost wanted to doubt you’d been my teacher. Interesting to hear how he knows you. But speaking of number theory, there’s this guy in my math class. I almost never see him because he refuses to come to class except for tests.”
“Where’s he from?’
“I’m not sure. He went to school in someplace called Lawrenceville, New Jersey, but it was a boarding school, I think, so I’m not sure where he’s from. Had a really good math program in high school and can’t stand our professor. Calls himself Stoney on account of his name, he says.”
“How’s he doing?”
“Only one test so far. He and I had the only 100s. The prof is plainly irritated by this. Not by me. I come to class and all. But he really hates Stoney.”
“Congratulations on the 100, Henry.”
“This is stuff you had me doing junior year, but thanks. Anyway, so Stoney wants to start a math club.”
“To do what?”
“He knows these other two guys who are interested in math and he wants to start a group to take on something more challenging.” I could hear her smoking and thinking about this.
“What would you do?” My turn to shrug.
“Something hard,” I said.
“Like what?”
“The only thing he mentioned was to see if we could deduce Kepler’s laws from Tycho’s observations.”
“Oh, gack, no. Henry, don’t join.”
“Why?” Unlike her to discourage an interest in Math.
“They were both crazy as coots,” she said. “Tycho was this nobleman with a common-law wife and he had all these extremely precise measurements of planetary positions because he was rich and could spend huge amounts of money on assistants and astronomical instruments. But he had no math whatsoever. He was a nobleman. Practical skills were beneath him.”
“So?”
“Tycho’s theory was that the sun and the moon and the stars revolved around the earth, and that the five planets— ”
“Five?”
“Mercury, Venus, Mars, Jupiter, and Saturn were it, back then. Uranus and Neptune weren’t discovered until there were really good telescopes, and Tycho didn’t have a telescope. Pluto was deduced from wobbles in the orbits of other planets, and Earth wasn’t regarded as a planet back then”
“Why is Pluto a planet? It’s not like the others,” I said.
“It’s out there. You leave Pluto alone.”
“Okay, Then if Earth wasn’t a planet, what was it?”
“The center of the universe.”
“They hadn’t read Copernicus?” I asked.
“Sure, they read it, but nobody believed him. Copernicus disagreed with what they’d been taught, and what they’d been taught explained everything pretty well. Plus, Tycho was really, really egotistical. He wanted his theory to be the one that explained the universe. He thought the Earth was the center of the universe, that the sun revolved around the Earth, and all the other planets revolved around the sun.”
“That’s insane.”
“No it’s not. If the universe were constructed like that, it would look exactly like it does now, for all intents and purposes. Think it through, later on. Imagine the Earth is stationary, and everything else is the way Tycho thought. It’s more complicated to imagine, but it would look exactly the same. It may actually be that way—there’s no way to know. But, anyway, Tycho had all these amazingly precise observations, but he had no math. No way to analyze all that data. So he brings Kepler from Austria to Prague, thinks of him as a hireling, and won’t share his data with him. Not really. He shows him some of his Mars stuff, because he’s convinced they support his inane Earth-Sun deal. Then Tycho dies, and his heirs are worse even than Tycho was. It’s all a mess.”
“But didn’t Kepler deduce the laws of planetary motion from Tycho’s observations?” I asked.
“Sort of.”
“Sort of?”
“’Deduce’ isn’t the right word. ‘Stumbled onto’ is better. Once Kepler at long last got access to Tycho’s log books—Tycho’s dead now, remember—he made some really clever deductions. But the only reason he did was that he made a huge calculation error in the orbit of Mars. Because of that he went on and recalculated Mars’ orbital path a number of times, and eventually realized that it wasn’t circular. But if he hadn’t made a huge error in the first calculation, he wouldn’t have kept calculating. All of Kepler’s laws came from that mistake, so it’s a good mistake.”
“So why do you think we should avoid Kepler and Tycho as a project?” I asked.
“They were all nuts,” she said. “Kepler thought his mother was a witch. Tycho complained that the King of Denmark didn’t value his services enough, and complained that the emperor or whatever he was in Prague wasn’t providing him a grand enough castle. They both cast horoscopes and believed in them.”
“Crazy people sometimes have great ideas,” I said.
“More likely they don’t,” she answered. “Most crazy people have crazy ideas. But the reason for avoiding a club that wants to deduce Kepler from Tycho is that it would be extraordinarily boring. You wouldn’t recognize any of Kepler’s calculations as Math.”
“What were they?” I asked.
“Arithmetic,” she answered. “Liebnitz and Newton and the calculus you like so much came in the next century. Kepler just sat in his little apartment in Prague and re-did the calculations over and over and over until they came out right.”
“πr2 all the way?” I asked.
“Yes. Each guess took thousands of calculations. All multiplication and addition. No functions at all. A complete pain in the … neck. You don’t want to spend time on that kind of thing. Months and months of rote calculation.”
“Okay. So , assuming there’s a club, what should we do instead, then?”
“Maybe something with Maxwell.”
“Okay,” I said, hesitantly.
“You don’t like John Clerk Maxwell?” she asked.
“Wasn’t he a physicist?” I asked.
“Yes, of course.”
“Well, the rest of these guys are math guys. I may be the only physics guy.”
“Then maybe the Lorentz transformations,” she said.
“What’s that?”
“Hendrick Lorentz described a set of functions (math functions that cannot be priduced in blogspot here, if you want a copy, send me a note)
where
(a short but complex equation should be here; can't insert either objects or pictures as text),” she said. “As best I can recall.” I thought that through for a moment.
“Leaving y-prime and z-prime alone for now, that would move around a lot,” I said, after a few seconds.
“No kidding.”
“I’m having a hard time imagining the graph. It must fold back on itself a lot.”
“It moves. It changes from 45 degrees off the x axis to 45 degrees off the y axis and then back again.”
“Cool. Is it good for anything?”
“Lorentz came up with it as a way to explain the propagation of light through the luminous æther.”
“The what?”
“The luminous æther. Going back to Aristotle, everybody assumed there was something that filled space, in a way. You’ve heard of fire, air, water, and earth. Aristotle said there was a fifth element that gave definition to space. In the nineteenth century there were a lot of discoveries that indicated that light was a wave of some sort, so a lot of physicists decided there must be some sort of medium the waves were passing through.”
“Why does there have to be a medium?” I asked.
“Now you’re thinking like Maxwell,” she said.
“Why?”
“Never mind. Up until Heisenberg and Bohr and those guys, we always thought about Physics in terms of physical realities. That’s why it’s a called Physics.”
“Oddly enough, a beautiful but crazy girl said that to me recently,” I said.
“Oh, yeah? Are you interested in her?”
“No, no. To me the crazy comes through much more loudly than the beautiful,” I answered. “So why does there have to be a medium?”
“Well, they thought the closest analogy to the way light moves was the way sound moves through the air, and they though that was because of regular compressions moving through the air. The compressions graph out to be a wave, even though they’re something else. If light was a wave, there had to be something for it to be compressing. A wave seems to need something to be moving through. So they referred back to Aristotle and called the stuff æther.”
“All interesting. Fascinating, in fact,” I said. “You are ever the teacher. So why are Lorentz’ calculations important? I’m thinking people probably don’t still believe in the æther.”
“No, but people did into the Twenties. Albert saw the Lorentz transformations as an accurate description of a misunderstood process and deduced Special Relativity from them.”
“Albert Einstein?”
“Yes,” she said. I could her exhale smoke.”
“I’ll suggest it to the guys but I’m the only physics student in the proposed club. There seems to be this odd hostility between mathematics and physics.”
“Well, that comes and goes. Look, Henry, I’m sorry, but I need to get off the phone. Are you set for money?”
“Oh, sure.”
“Are you coming home for Thanksgiving?”
“Geez. I hadn’t thought about it. I don’t know where my parents or my sister are.”
“No, Henry, I meant are you coming to home Chattanooga? Do you want to have Thanksgiving with us?”
“Well, sure, thanks. I don’t have a car, though.”
“So?”
“I guess I can take a cab from the Green Ghetto to your house. And I can take the bus from Nashville to Chattanooga.”
“Henry, for Heaven’s sake. Don’t make it complicated. You can stay with me. Call Ginny and you two come down together. I’m sorry. I really have to run.”
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