Exploring Zen and Math: The Marvelous Arrows of Category Theory
I continue my journey through mathematics and Zen Koans
"The Zen masters - ancient Japanese sages who still teach us to see life new from their temples -" I say softly as the rain taps on the glass panes, making small beats that match my voice. The mix of chalk scent and fresh air flows in from the small gap in the glass. "Now I'll share a good tale," I say, as I draw a big ring on the board. "Back then, in a temple, a wise man asked his small child: 'How does one hand sound when it claps?'"
Sara shook her head fast: "That's wrong! You must use both hands to clap!"
Tommaso, from the back of the classroom: "It's just impossible."
"I don't understand," Lisa mumbles, frowning.
"These words sound strange, don't they?" I smile. "Watch this." I raise one hand and move it through the air. "What sound does it make? And now..." I take the chalk and draw a line connecting the circle to another one. "This line is like the movement of my hand. It makes no sound, but it creates something."
Marco leans forward from his desk: "My cat! He walks and you can't hear anything, but he moves!"
"And sometimes he leaves footprints!" adds Sofia, jumping in her chair.
"Beautiful! You see? The silent cat leaves traces, and creates connections." I draw more circles on the blackboard and connect them with arrows.
"These circles are like the soap bubbles you make in the garden. But the magic isn't in the bubbles... it's in the arrows that connect them! Just like the silent sound of the hand moving through the air."
Anna raises her finger: "Numbers! One comes before two, then comes three!"
"Excellent! Between one and two there's an invisible arrow. And between two and three. And you know something even more magical? Mathematicians have discovered that the whole world is made of these arrows!"
I turn to the blackboard and draw:
A circle with "1" and a circle with "2", connected by a "+" arrow
A circle with "seeds" and one with "flowers", connected by a "growth" arrow
A circle with "silence" and one with "sound", connected by a "movement" arrow
"The koan tells us that silence is important too, just like these arrows that connect everything."
"Teacher, what's a koan?" asks Andrea, timidly raising his hand.
"Ah, right!" I smile. "A koan is like a special riddle that Zen masters used to make their students think differently. Like when I asked you about the sound of one hand clapping. It seems nonsensical, right? But it makes us see things in a new way, just like our magic arrows!"
Lisa holds up her pencil case: "I get it! It's like my markers... each one makes a different color but together they make the drawing!"
"Perfect! And if we do 2+2, the arrow takes us to 4. It's truly a magic bridge between numbers."
I distribute graph paper. "Let's play a math game. Draw two circles with numbers and connect them with an arrow that shows what happens when you put them together."
Marco, who was confused earlier, now draws enthusiastically: "I put 3 and 6... the arrow says the second is double!"
"I did 5 and 10," whispers Sofia. "My arrow says plus five."
The bell rings. Piero, always so quiet in the last row, springs up like a jack-in-the-box. His chair falls backwards with a sharp thud. His cheeks turn bright red as he rushes toward the door, backpack clutched to his chest.
"Well," chuckles Marco, "he's not like my cat... you can definitely hear him move!"
The laughter spreads through the classroom like a wave, warm and gentle. I join in too, while Piero, already in the hallway, hints at a shy smile.
I linger for a moment at the window, watching the raindrops continue to fall. Two tiny drops slide close together on the glass, separated by a paper-thin space, almost invisible. Like the arrows on our blackboard, I think, even that space tells a story. It's a transparent bridge, a silent connection. I move even closer to the glass until I see the world through that tiny distance between the drops, and for an instant, I seem to deeply feel what the Zen master meant by the sound of one hand clapping. Echoing softly in the distance, the final, muffled notes of Kenny G’s ‘I Can’t Tell You Why’ drift through the air.
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