"Master Leibniz, isn't this just a simple math problem?" Tiek asked, puzzled.
Even apprentices could solve it, let alone wizards proficient in arcane mathematics.
Alva and the others were equally disappointed. Was this the puzzle tornting the entire realm of arcane math? Was it?
"Do you truly find it simple?" Leibniz looked at everyone, expressing regret. "The issue lies not in when they can catch up, but in why they can catch up."
"Zeno told , at his speed, it takes ten seconds to reach the starting point of the turtle! But by the ti he arrives, the turtle has moved a ter. Though the distance between them has closed significantly, there's still a ter gap. So, he needs another tenth of a second to reach the turtle's current position. Yet, by then, the turtle has moved again, requiring him to spend a thousandth of a second catching up to the new position"
As Leibniz explained, he extended his right hand, drawing a magical line in the air, denoting the start and finish of the race. He used red light to mark Zeno's progress and green light for the turtle's. Despite closing in, a minuscule length persisted between them, an infinitesimal yet persistent gap.
Zeno dashed forward, seemingly unable to catch up to the leisurely turtle before him.
Tiek and the others stood stupefied, their expressions shifting from confusion to gravity, quickly descending into contemplation.
The theory was easy to comprehend: the wizard nad Zeno, in pursuit of the turtle, inevitably passed the creature's starting point. But when he arrived at this point, the turtle had crawled forward, creating a new starting point, leading to an endless cycle of deduction...
Alva pondered deeply, sensing sothing amiss but unable to pinpoint it precisely.
He was unaware that this contradiction was a clash between reality and logical mathematical deduction.
Tiek was nearly dizzy from the ntal gymnastics. It took him a while before he suddenly grasped sothing. "Wait, Master Leibniz, no matter what, at the eleventh second, Zeno should always catch up to the turtle, right?"
"That's precisely the problem, my friends!" Leibniz nodded, then emphasized, "If ti and space are infinite and infinitely divisible, logically, the later participant in a race can never surpass the forr, as they're separated by an infinite number of fractions."
"This distance, in a sense, is infinitely long, for it can be divided into countless fractions!"
"But if Zeno can inevitably catch up to the turtle, does that not imply that in our world, space and ti are not continuous but possess a minimum scale? It's because Zeno, as the later participant, crosses this smallest scale at so point, allowing him to catch up to the leading turtle..."
"Your contemplation is truly thought-provoking, Master Leibniz!" Alva breathed out, expressing admiration.
Only now did the wizards understand that the debate between these two masters of arcane mathematics wasn't truly about a re racing problem; it delved into whether a value could be infinitely subdivided and probed the existence of the smallest scale in ti and space.
"So, you've reached a conclusion and won this dispute, haven't you?" Tiek exclaid, fascinated by the deductive leap from an inevitable winning race to the potential existence of the smallest scales in ti and spacea truly creative line of thought that earned his admiration!
"Not quite, for if that were the case, I wouldn't be able to answer his second question!" Leibniz lanted.
There was another question? Alva and the others felt a chill run down their spines.
Leibniz extended his hand, summoning an iron arrow into the void, which swiftly embedded itself into a nearby bookshelf. He turned to the group and asked, "Do you think this arrow, once shot, has moved or remained still?"
Another seemingly simple question that left Tiek, Ellison, and the others pondering for a long ti, contemplating if there might be a deeper aning hidden within.
Alva, on the other hand, didn't dwell much. He decisively stated, "It has moved, of course!"
He had seen it with his own eyes, and no eloquence could change that fact!
"If, according to what we just discussed, ti exists in the smallest scale, does this an that in each of these smallest scales, the arrow has a definite position, occupying the sa space as its volu?" Leibniz continued.
Alva furrowed his brows, contemplating for a while before cautiously stating, "I believe so."
"So, disregarding other factors, in that mont, is the arrow moving or still?" Leibniz pressed on.
"Undoubtedly still!" Alva firmly responded.
Tiek and the others nodded, easily envisioning a suspended iron arrow if ti were to halt at a certain point.
"If this mont is motionless, what about other monts?"
"Those should... also be motionless?" Alva responded uncertainly.
"In other words, at each point in ti, it remains stationary. So, the arrow shot is also motionless, correct?" Leibniz concluded.
"Of course..." Alva hesitated in his reply, then froze entirely. How could a flying arrow be motionless?
Tiek, Ellison, and others frowned deeply.
If Leibniz's earlier statent was correct, that ti existed in the smallest scale and was indivisible, then following the logic, each mont of the iron arrow was motionless. Hence, the flying arrow couldn't be in motion. After all, how could sothing constantly motionless be called in motion?
Could it be that an infinite sum of stationary positions equaled motion itself? Or perhaps, infinite repetitions of stillness constituted motion?
If Leibniz's statent was wrong, and there was no such thing as the smallest scale, if ti could infinitely subdivide, and everything was continuous, then the flying arrow would naturally remain in motion. This ford the basis for the paradox's dissolution.
But if that were true, would Zeno never surpass the turtle?
The assembled individuals suddenly felt themselves swirling in a colossal vortex, teetering between the motion and stillness of the iron arrow, Zeno's potential catch-up with the turtle, their minds on the verge of collapse...
Leibniz observed Tiek and the others lost in contemplation and couldn't help but smile. These two paradoxes, simple as they appeared, would have sparked the second mathematical crisis if placed in the 17th or 18th century!
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