"Master Leibniz, isn’t this just a simple arithtic problem?" Tic asked, perplexed.
Not to ntion wizards proficient in advanced mathematics, even an apprentice could solve it.
Alva and the others were extrely disappointed, was this the problem that baffled the entire community of advanced mathematics? Just this? Just this?
"Do you really think it’s simple?" Leibniz looked at the people present and said regretfully, "The problem is not about when he will catch up, but why he can catch up."
"Zeno told that at his speed, it would take ten seconds to reach the starting point of the tortoise!
But by the ti he got there, the tortoise had already moved one ter. Although the distance between them had closed considerably, there was still a one-ter gap. So, he needed another tenth of a second to reach the tortoise’s new position. Yet by that mont, the tortoise had moved another distance, so he must spend one-thousandth of a second to catch up to the tortoise’s position…"
As Leibniz spoke, he extended his right hand and drew a line in the air with magic power, indicating the start and finish of the racetrack. Then he used red light to show the distance Zeno had covered and green light to show the distance the tortoise had travelled; the two distances kept narrowing, but there was always a slight length separating them, this distance, no matter how small, still existed...
The sprinting Zeno seed unable to catch up with the slow-moving tortoise no matter what...
Tic and the others stood in stunned silence, their expressions slowly turning solemn. Soon, they fell into deep thought.
This theory was easy to understand: during the chase, the wizard nad Zeno inevitably had to pass through the starting point of the tortoise, but every ti he reached that point, the tortoise would have crawled forward a bit, aning that there was a new starting point waiting for him, this could be infinitely iterated...
Alva thought hard and ditated, feeling sothing was off, but couldn’t figure out exactly where the problem lay.
He didn’t realize this was the contradiction between reality and the deduction of mathematical logic.
Tic was almost dizzy from the circular reasoning and took a while before he suddenly caught on. "Wait, Master Leibniz, no matter what, by the eleventh second, Zeno could always catch up with the tortoise, couldn’t he?"
"That’s precisely the issue, my friends!" Leibniz nodded and then stressed his tone sowhat. "If ti and space are infinite and can be continuously divided, then logically, the one behind in a race could never overtake the one ahead, because there is an infinite number of one percent gaps between them.
This distance is, in a sense, infinitely long, after all, it can be divided into countless equal parts!"
"But if Zeno must catch up with the tortoise, does that an that in our world, space and ti are not continuous, but there exists a minimum scale for both space and ti? It was because Zeno, as the one behind, crossed this minimum scale at so mont that he was able to catch up with the tortoise ahead…"
"Your thoughts are truly insightful, Master Leibniz!" Alva exhaled and said admiringly.
Only then did the group of wizards realize that the two masters of advanced mathematics were not truly entangled in a so-called racing problem; the crux of the debate was whether a value could be infinitely divided, probing whether there is a minimum scale for ti and space.
"So you’re saying, you have already co to a conclusion, securing the victory in this dispute, haven’t you?" Tic spoke cheerfully, using an inevitably victorious race to infer the possible existence of the smallest scales of ti and space, this creative way of thinking genuinely impressed him!
"Not at all, because if that were true, then I couldn’t answer his second question!" Leibniz said, quite distressed.
There’s a second question? Alva and the others suddenly felt a tinge of panic.
Leibniz extended his hand, and an iron arrow appeared in the void, striking a bookshelf at a rapid speed. Then he turned to look at the others and asked.
"What do you think, has the arrow that was shot moved, or has it not moved?"
Another simple question that didn’t seem to require any thought to answer, yet Tic, Ellison, and the others hesitated for a long ti, wondering if there might be so deeper aning.
Alva, on the other hand, had no such scruples and said decisively, "Of course it moved!"
He had witnessed it with his own eyes, right in front of him; no matter how much the other side argued, it wouldn’t change this fact!
"According to what we just discussed, if ti has a minimum scale, then in each minimum scale does this iron arrow occupy a definite position, and does the space it occupies match its volu?" Leibniz continued to inquire.
Alva frowned and pondered for a long ti before he replied cautiously, "I think so."
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"Then, not considering other factors, in this instant, is the arrow moving, or is it not moving?" Leibniz followed up.
"Of course it’s not moving!" Alva answered with certainty.
Tic and the others also nodded, noting that if one conceives of ti stopping at a certain mont, then naturally one could see a stationary iron arrow.
"If this instant is not moving, what about other instants?"
"They should… also be not moving?" Alva said with uncertainty.
"So you’re saying at every mont it is stationary, then the arrow that was fired is also not moving, right?" Leibniz asked finally.
"Of course…" Alva hesitated and replied, then his whole being froze. How could an arrow in flight be not moving?
Tic, Ellison, and the others all furrowed their brows.
If what Leibniz said before was correct, that ti has a minimum scale and cannot be further divided, then following the logic just deduced, in each instant the iron arrow is stationary, then the arrow that flew out cannot be moving, because how can sothing that is always stationary be said to have moved?
Could the sum of an infinite number of stationary positions equal motion itself? Or could endless repetition of stillness be motion?
If Leibniz’s statent was wrong, and there is no such thing as a minimum scale, and ti could be infinitely divided, everything being continuous, then the flying arrow would naturally always be in motion, and this paradox wouldn’t exist.
But if that were so, wouldn’t Zeno never be able to surpass the tortoise?
The people present suddenly felt as if they were caught in a giant vortex, wavering between the motion and stillness of the iron arrow, the paradox of whether Zeno could catch the tortoise, their brains felt like they were about to burst…
Leibniz watched Tic and the others deeply engrossed in thought and couldn’t help but smile. These two paradoxes might seem simple, but if put into the seventeenth or eighteenth century, they could trigger a second mathematical crisis!
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