(With apologies to Zeno of Elea)

This story is based on a philosophical conundrum proposed by Zeno of Elea, an ancient Greek philosopher. I first heard about it whilst studying Latin at school, and felt it would make an interesting topic for a short story. In fact the story is not so much about the conundrum, as it is about the conflict between the physical world and the mental one, and about how much faith we can place in what we are told.

One day I might even complement this one with a piece about my favourite philosophical conundrum, which quite convincingly proves that the impact of a fly on a fast moving train causes the train to stop dead.

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It was a freezing winter's day. The trees were frosted with a light layer of snow, and the path was mottled by a crazy dance of whiteness where the foliage above allowed individual flakes to pass through. The ground was as hard as diamond, preserving exactly the footprints of the travellers that had passed during the autumn. The tracks of their cartwheels left tramlines which were now fixed into place, and lined with razor sharp edges of splintered ice.

The warrior stepped onto one such edge, feeling it take the pressure well until he lifted his other foot from the ground, and felt the shattered peak crunch satisfyingly down beneath him. Walking in fairy steps he repeated the movement again and again, moving deeper into the forest with every move.

Eventually the time came for him to leave the tramlines and walk the bare soil of a footpath which veered to the north. He knew this path well, having travelled it many times before, but the translucent whiteness that covered everything made it as fresh to his eyes as the first time he'd walked it. The path turned a corner, and there it was; the bridge.

It was hardly spectacular - just an old oak tree, felled and trimmed to make a crossing point. The feet of a thousand travellers had worn it smooth, and the sheen of ice that glinted upon it reminded him that he would have to be careful as he crossed. Beneath the bridge, just a few feet below, ran the icy waters of the forest stream. It was fairly narrow, and easy enough to cross by foot - but in this weather it wasn't prudent to get any wetter than was absolutely necessary.

He walked the soil bank which ramped up to the bridge's top, and stepped on. As he placed more and more weight onto his leg, his body confirmed that it was slippery underfoot, and that he would have to take every step with great care. He trod slowly, watching his feet as he placed them down square to the wood, and assuring himself of his footing at every step. Just ten yards of this, and he could continue his journey with relative ease.

"Back up would you, there's a good man."

Ensuring his feet were planted securely, the warrior lifted his head. There, at the other end of the bridge, was a well dressed man just stepping up onto the log.

"Wait there will you,there's not enough room on the bridge for us to pass each other."

"That's why I asked you to back up. Come along now, I've got better things to do than chat to you all day."

"But I was on the bridge first."

"Yes, yes. But I'm here now, and I want to cross."

"I don't care. I was already here, and I don't intend to back up for anyone."

"Is that so?" The man took a step forwards.

"Furthermore, I am prepared to defend my position by force if necessary."

"So on a slippery bridge you want to swing your sword around, and send us both into the stream. Nice in principle, but not so hot in practise, so why don't you just back up and let me cross - save yourself getting too wet."

The warrior hadn't really thought of the consequences of his statement, but had no intention of withrawing the challenge. "What I mean is that if you continue to make your way across the bridge, rather than waiting until I've finished crossing as any sensible man would, I shall have no choice but to loose an arrow at you." He folded his arms in defiance.

The man took a step forwards.

"I'm warning you..."

Another step. The warrior drew his bow.

"I don't want to hurt you, but I am prepared to use force if necessary."

The man took another step forwards. The warrior notched an arrow.

"This is your last chance. One more step and I fire."

The man stepped forwards once again. The warrior pulled back on the fibre of his bow.

"I'm warning you - I will do it, you know."

Another step. The warrior released the string, sending an arrow racing towards the man. It span as it flew, screwing its head through the air and flying true to the warrior's aim. In the blink of an eye it was within inches of the man, where it stopped dead, hung in the air for a second, then fell straight down, bouncing off the wood of the bridge and splashing into the stream below.

"What sort of sorcery is this?!" The warrior quickly notched, drew and released another arrow. It span towards the man, true to the warrior's aim, then stopped, hung in the air for a second, and dropped down to clatter off the log and splash into the water.

"No sorcery. Just simple laws of nature."

"What do you mean? That was magic if ever I've seen it."

"I'm afraid not. It was just common sense really - there's no way any of your arrows can ever hit me."

"Explain yourself. I've never missed a shot before and I want to know what kind of power you posess to make me start."

"You've never missed a shot before? Hmmm... have you ever stopped to think about the flight of your arrows?"

"Not really. I just know how hard to pull, and where to aim if it's windy."

"Okay, let me walk you through one of your shots, and you can see why you'll never hit me. Let's suppose you were at one end of the bridge, and I was at the other. You let an arrow go and it starts to fly towards me."

"Right."

"So your arrow's happily flying along, making its' way towards me, when it reaches the halfway point. So it's still got half the bridge to go before it reaches me. In fact, it's just as though you'd tried to shoot from a bridge that was half as long."

"Okay..."

"It carries on, but eventually it reaches another halfway point. This time it's halfway between the real halfway point and the end of the bridge - so it's three quarters of the way along in real terms. What that means, though, is that it's still got a quarter of the way to go. Then it reaches a point halfway between that last point and the end of the bridge. Now it's got an eighth of the way left to go. Then it flies on until it's got a sixteenth of the way to go, then one thirty-second part, then one sixty- fourth part."

"Okay, I see that."

"Then you should also see that every time it reaches one of those points, it still has further to go before it reaches me. No matter how long it's been flying, no matter how close it gets, it'll always have one hundred-and-twenty-eighth part left to go, or one four-thousand-and-ninety- sixth part. It'll always have some distance left to travel, and so it never actually reaches me."

"But that doesn't make any sense - I've shot plenty of people before."

"But not any that actually understood what you were doing. You've seen for yourself that your arrows don't hit me."

"I don't care - I'm still not budging from this spot."

"That's alright - you couldn't if you tried."

"What d'you mean?"

"Well, as I've just explained to you, your arrows can never reach their destination."

"So what's that got to do with me moving?"

"Well, by the same mechanism, you can never reach your destination either."

"Maybe not, but I can still move."

"Oh no you can't. Just imagine; if you wanted to cross this bridge, you'd have to pass the halfway point, just like your arrow did."

"Yeah."

"Which means that for the first half of your journey, the halfway point is your destination."

"And...?"

"And, as I've already proven, you can't reach your destination."

"Okay, so I can't reach the halfway point of the bridge - but I can still move."

"It's not just that you can't reach the halfway point - you can't reach a point one quarter of the way along, either. Or one eighth. Or one sixteenth. In fact, we could carry on like that until we get to a distance that is so small that you can't measure it, but if you want to cross the bridge, then that point id your first destination. And you can't reach your destination."

"I see what you mean."

"Shhh... You can't talk either, remember. In order to talk, you have to move your mouth, and if you can't even take a step forwards then you certainly won't be able to do that."

The warrior was silent.

"In fact, you shouldn't really be breathing. I mean, if your foot can't move a hair's breadth forward, then your chest can't rise and fall either."

The warrior thought for a few moments, then stopped breathing. A look of panic almost came over his face, until he realised that he couldn't even move his facial muscles. The man looked on as the warrior turned blue, and fell from the bridge into the icy waters below. He made his way across the bridge, stopping only once, at the point where the warrior had stood. He looked over the edge to the body below.

"That'll teach you to pick a fight with a philosopher."

©1997 Xav

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