"Master Leibniz, isn't this just a simple math problem?" Tike 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 tormenting 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 me, at his speed, it takes ten seconds to reach the starting point of the turtle. But by the time he arrives, the turtle has moved a meter. Though the distance between them has closed significantly, there's still a meter 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 a red light to mark Zeno's progress and a 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.
Tike and the others stood stupefied, their expressions shifting from confusion to gravity, quickly descending into contemplation.
The theory was easy to comprehend: the wizard named 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 something amiss but unable to pinpoint it precisely.
He was unaware that this contradiction was a clash between reality and logical-mathematical deduction.
Tike was nearly dizzy from the mental gymnastics. It took him a while before he suddenly grasped something. "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 time and space are infinite and infinitely divisible, logically, the later participant in a race can never surpass the former, 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 time are not continuous but possess a minimum scale? It's because Zeno, as the later participant, crosses this smallest scale at some 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 mere racing problem; it delved into whether a value could be infinitely subdivided and probed the existence of the smallest scale in time and space.
"So, you've reached a conclusion and won this dispute, haven't you?" Tike exclaimed, fascinated by the deductive leap from an inevitable winning race to the potential existence of the smallest scales in time and space—a 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 lamented.
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 Tike, Ellison, and the others pondering for a long time, contemplating if there might be a deeper meaning 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, time exists in the smallest scale, does this mean that in each of these smallest scales, the arrow has a definite position, occupying the same space as its volume?" Leibniz continued.
Alva furrowed his brows, contemplating for a while before cautiously stating, "I believe so."
"So, disregarding other factors, at that moment, is the arrow moving or still?" Leibniz pressed on.
"Undoubtedly still." Alva firmly responded.
Tike and the others nodded, easily envisioning a suspended iron arrow if time were to halt at a certain point.
"If this moment is motionless, what about other moments?"
"Those should... also be motionless?" Alva responded uncertainly.
"In other words, at each point in time, 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?
Tike, Ellison, and others frowned deeply.
If Leibniz's earlier statement was correct, that time existed in the smallest scale and was indivisible, then following the logic, each moment of the iron arrow was motionless. Hence, the flying arrow couldn't be in motion. After all, how could something constantly motionless be called in motion?
Could it be that an infinite sum of stationary positions equalled motion itself? Or perhaps, infinite repetitions of stillness constituted motion?
If Leibniz's statement was wrong, and there was no such thing as the smallest scale, if time could infinitely subdivide, and everything was continuous, then the flying arrow would naturally remain in motion. This formed 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 Tike 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.
TL/n -
Zeno of Elea (c. 490 – c. 430 BC) was a pre-Socratic Greek philosopher from Elea, in Southern Italy (Magna Graecia). He was a student of Parmenides and one of the Eleatics. Zeno defended his instructor's belief in monism, the idea that only one single entity exists that makes up all of reality.
He rejected the existence of space, time, and motion. To disprove these concepts, he developed a series of paradoxes to demonstrate why they are impossible.
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Zeno's arguments against motion contrast the actual phenomena of happenings and experiences with the way that they are described and perceived. The exact wording of these arguments has been lost, but descriptions of them survive through Aristotle in his Physics
en.wikipedia.org/wiki/Zeno_of_Elea