Bryony had decided that she did not have the temperament for being late or for being lost. In consequence, she was the first one into the Atkinson lecture theatre. She watched the rest of the class trickle in from breakfast.
Melinda and Buhle were talking excitedly as they arrived. “Mike must’ve put the posters up last night. I’m sure they weren’t there yesterday. And it’s barely a week’s warning!”
“Well, a week seems like enough warning for an audition, especially since we’ve been expecting it anyway. You worry too much.” Buhle shook her head.
“But worrying is half the fun. It’s not like there’s much else we can actually do at this point.”
“What’s Melinda worrying about so happily?” Bryony asked.
“The robo-duelling auditions posters went up last night,” Buhle said.
“Really?” Kelly Jean had just walked in. “Why did nobody tell me? I still need to prepare!”
“It’s hardly our responsibility to keep you updated if you can’t read the news bulletins for yourself, Kelly Jean,” Bryony said. She rolled her eyes, wondering why Kelly Jean made everything about Kelly Jean.
“We were going to tell everyone,” Buhle added, “but Melinda only saw the posters on the dining hall noticeboard after breakfast. It’s not like we’ve had much chance.”
There was an awkward silence, broken by a group of new arrivals. “Hey, have you guys seen that the robo-duels audition announcement is out?” Ken asked.
“Oh dear stars and planets.” Bryony buried her face in her hands. “Is that all anybody can talk about?”
“It’s exciting for those of us taking part, silly.” Melinda sat down next to her. “You should come watch.”
Bryony snorted. “No thanks. I hope you guys have fun, but I somehow doubt your auditions will be anything as exciting as the City Hall matches.”
“Fair enough.” Melinda grinned and leaned down to grab her notebook as Mathematician Liang walked in at the front of the lecture theatre.
Mathematician Liang fiddled with the viewing screen for a few minutes. Once she had connected it to her notebook, a pair of graphs appeared and she turned back to the class. “Good morning, all of you. Today we will begin with the real substance of what we might call Real Analysis. So far, we have had a very hand waving overview.” She gestured dismissively. “I hope that the overview and your afternoon calculation exercises have helped you to become comfortable with these ideas of differentiation and so on. Today we will begin to properly define things, so that we can go on to proving the theorems we want.
“The graphs on the screen should be familiar to you. If we want to look at the difference between two points, we can think about connecting them with a line that forms a secant. If we bring those points closer and closer together, it seems intuitive that the secant will eventually become a tangent, so that it touches the curve at only one point. This is when the two points have become equal. Now, that’s a pleasant enough intuitive picture, but merely expecting something to happen is not particularly useful if we want to prove how things will actually behave. Perhaps you will offer me some thoughts on this.” She stopped and looked out over the class.
There was silence for a few moments and then Buhle said thoughtfully, “I think there was a thing on Ancient Earth when they didn’t really believe in gravity. I read somewhere that even after people made measurements, it was sort of controversial that objects would accelerate as they fell, because it wasn’t what people expected. I’m not sure if that’s exactly the same as a proof, though.”
Mathematician Liang smiled and nodded at Buhle. “We can relate the ideas, certainly. In pure mathematics we need proofs to make sure that what we assert is consistent with our starting definitions. In physics the most important thing is to be consistent with measurement. Either way, what is consistent is not necessarily what you would expect.
Melinda said, “Once when I was little I thought I’d be able to double a number and get an odd answer. Of course, we know you can’t now, but I spent ages trying to do it before my brother showed me why you can’t. I think my expectations are a bit more sophisticated now, but the same idea still applies, I guess.”
It was the kind of thing Melinda would do, Bryony thought. She was always tinkering with numbers and patterns without any particular aim.
“Very nice, Melinda. I think you will all find that your expectations are challenged as you begin your studies of the Calculus. The infinite and the instantaneous are both quite outside of the everyday.”
The exercises they’d done so far hadn’t surprised Bryony, but Mathematician Liang had said that was the intuitive stuff. She wasn’t sure what infinity had to do with anything.
“Now, let’s move on to some definitions.” She turned a page in her notebook and the screen displayed, “Let ε > 0.”
“We’ll be using the symbol epsilon to represent a small, positive quantity throughout this course, as mathematicians have for centuries. Now, can epsilon equal zero?” Mathematician Liang turned to peer at Bryony.
“Uh, no, it’s greater than zero, right?”
“Good. Remember that in the weeks to come. We can make epsilon arbitrarily small, so that it comes as close to zero as you like, but from this definition, epsilon will never equal zero. Sometimes that turns out to be important. For instance, if we work out the slope of the line between two points, we divide by the distance between the points, yes?” The class nodded. “Now, in your exercises, you have been working out the slope of a tangent, which is the limit where the two points fall on top of one another.”
Bryony hadn’t thought about that. If the two points were actually the same point, dividing through by their difference would mean dividing by zero, which didn’t seem like a good idea.
“Some of you have already expressed some worry about this.” Mathematician Liang smiled. “That is an excellent attitude to cultivate. As your expectations become more sophisticated, you will become more aware of such issues. Epsilon greater than zero is a tool we will use to solve many of these inconsistencies.
“Now, one more question for you before I move on to more complicated definitions.”
Bryony hadn’t really thought “ε > 0” was a definition, but it did seem to be all the information they needed.
“If the absolute value of x is less than epsilon and we allow epsilon to have any value greater than zero, what can we conclude about x? I hope you remember that taking the absolute value just means throwing away the minus sign if there is one.”
Bryony wasn’t sorry for the reminder. That meant x should be a positive number that was smaller than any other positive number. No, that didn’t make sense, since you could divide any positive number by two and get a smaller positive number. Bryony wasn’t sure Mathematician Liang’s question made sense.
“It means x must be zero,” Quintessa said. “Otherwise you would be allowed to make epsilon equal to the absolute value of x divided by two, which would break the inequality.”
“Very good, Quintessa. Now, are all of you happy with this result? Quintessa has summed up the argument well.”
Bryony felt like she was squinting to make sense of it, but Quintessa’s logic was solid. The only way everything could be true at once was if x was zero. She guessed that was what Mathematician Liang meant about proofs making sure everything was consistent.
By the end of the lecture she had decided proving that the Calculus made sense was a lot harder than just using it. Her brain was tired. When Mathematician Liang had left, she stood up to stretch in an attempt at defogging her mind. She wasn’t the only one.
“That was great,” Melinda said.
“I think I’m going to be much happier about using all those rules once we prove them properly from the ground up,” Quintessa said.
“I was happy enough just using the rules, but it’s really fun seeing how to work them out properly like this.”
“You guys are crazy,” Bryony said as she came up from touching her toes. “I would be A-okay with just using the rules as long as somebody else has shown that they work.”
“I take it that Becky has begun her torture sessions.” Mathematician von Rejk had come in while they were talking.
Bryony nodded violently as Quintessa protested, “It was beautiful!”
Von Rejk laughed. “To each his own. There’s some use in understanding why the rules don’t always work, but I was as happy as the next engineer to drop Analysis in my second year. Daresay Becky feels the same about my sloppy modelling. Speaking of which-” He turned and began writing on the board.
Bryony sighed and sat down again. She was soon too absorbed in the lecture to worry about other classes.