The lesson began with the simple instruction, “I want you to think carefully before replying.” “If I said to you I am a liar-” I commenced.
“We know that,” Daniel interrupted with a grin. “My maths report proves it."
“Daniel, the lions are waiting in the arena,” I was goaded to reply. This brought appreciative laughter and cries of “Go, sir!”
“If I said to you I am a liar, am I telling the truth?"
“Yes, er, no, I mean yes, I think, maybe,” Francesca vacillated, furrowing her brows in concentration.
“Do you want to change your answers, Francesca?” I asked wryly. “Anyone else?"
Helda offered a critical response. “If you are a liar, then your statement ‘I am liar’ is not true. This makes ‘I am a liar’ to mean ‘I am not a liar’. Therefore you must be telling the truth."
A round of applause was an invitation for me to acknowledge the alleged success of Helda’s Socratic method. Wen, however, was pensive. He proposed a corollary to Helda’s analysis.
“But if he is telling the truth, then it was also true when he said ‘I am a liar’ and we’re back where we started."
At this point Jimmy forfeited his membership of the society of natural philosophers.
“I have a headache, sir. Truths are lies and lies are truths. Something’s wrong somewhere.” He rested his head resignedly on the desk.
“Exactly, Jimmy,” I agreed consolingly. “We are chasing our tail.”
I offered clarification. “What we have is a paradox; a statement that contradicts itself. Here’s one you may know. The irresistible force meets the immovable object."
“I know that one,” Stavros informed us. “It’s like your mum telling you to cut the lawn just when the Grand Final is starting on TV.”
“I’m not scared of my mum,” Chris boasted irrelevantly. He was rebuked by calls of ‘liar’ by his friends.
“This leads us to a series of paradoxes first proposed by the Greek philosopher Zeno, born in Italy about two and a half thousand years ago."
“Hey,” Jimmy protested emphatically as he raised his head from the desk, “How could he be Greek if he was born in Italy?"
I tried unsuccessfully to clarify the situation. “He was born in Southern Italy, which at the time was a Greek colony."
“Italy was Greece and Greece was Italy,” Jimmy wailed before lowering his head again onto the desk.
“That is a paradox in itself,” Janice spoke, believing her comment was epiphanic.
“Okay, I need an assistant,” I remarked. Steve’s hand was up first.
“Good. Steve, go to the corner. Now, I’m going to prove to all of you that Steve is incapable of walking to the other side of the room."
“Break his legs, sir,” Alex called out gratuitously.
“The room is about eight metres wide, right?” I asked, soliciting agreement. I waited for nodding heads before continuing.
“Before Steve can reach the other side, he must first reach the four metre half way point, correct?” More head shaking followed.
“But before he can reach half way, Steve has to get a quarter of the way, which is two metres from the start. Are you still with me?” I wanted to know.
This time heads were motionless but I knew brains were thinking.
“And to make it a quarter of the way through, Steve must first walk one eighth of the required distance, but before that he must get to one sixteenth of the way, and so on.”
I paused to catch my breath and to allow assimilation of the data.
“We are halving an infinite number of times,” I stated as I prepared for the punch line. “And since infinity never finishes, Steve will never be able to take that first step."
Steve pretended he was trying to break free from invisible leg shackles.
Yao encapsulated everyone’s puzzlement by exclaiming, “That’s weird!"
To allay her concerns I asked, “What is the flaw in Zeno’s method of bisection?”
“What does bisection mean?” Janice asked.
“Maybe they didn’t have fractions in ancient times,” Lino offered simplistically.
“It depends on the units they used,” was another naive hypothesis. In exasperation and to obtain absolution for my incompetent teaching I pleaded, “Helda, what do you think?”
“If you halve something an infinite number of times, sooner or later you will get to the end,” she argued convincingly. Her sophistry was persuasive but incorrect.
Jimmy raised his head and completed the puzzle. “When you add fractions together, you always get an answer."
“Precisely,” I responded. “Zeno assumed that the sum of the infinite sequence –in our case 4+2+1+0.5+0.25 and so on - is limitless, but in fact, it has a sum of 8, which is the width of the room."
To visualise the idea of a limit, I got students to stand on a length measured from one wall. Steve stood 4 metres from the wall, Helda was assigned 2 metres in front of Steve, and so on. The seventh person, Jimmy, took his position very close to his neighbour, and the eighth person stood no more than several centimetres from Jimmy.
The inevitable jostling created a snapshot reminiscent of the famous World War 2 photograph, Raising The Flag On Iwo Jima. Exhausted, the group slumped onto the carpeted floor and burst into convulsive fits of laughter and watched Mako gyrate rap style.
Engulfed by the proceedings, I did not notice that the Principal was ushering a group of parents into the room on a tour of the school. They stared with obvious bemusement.
“And what maths are we studying today?” the Principal asked as if this was all preordained.
“We’re demonstrating empirically how to locate the equilibrium point and centre of mass of rotating bodies using Cartesian nomenclature and integral calculus,” I blurted out without hesitation.
Readers may deliberate at their leisure on the aftermath. It was a lesson to remember.