Before you can reach your destination, you must first travel halfway. Before halfway, a quarter. Before a quarter, an eighth... You must complete INFINITE steps before taking even ONE!
To reach the goal, first travel ½, then ¼ more (reaching ¾), then ⅛ more (reaching ⅞)...
The series: ½ + ¼ + ⅛ + 1/16 + ...
Problem: You never actually ARRIVE—always ½ of the remaining distance left!
Before reaching ½, you must reach ¼. Before ¼, reach ⅛. Before ⅛, reach 1/16...
The series: ... + 1/16 + ⅛ + ¼ + ½
Problem: Infinite steps with NO FIRST STEP! How can you even begin?
"That which is in locomotion must arrive at the half-way stage before it arrives at the goal." — Aristotle, summarizing Zeno
If space is infinitely divisible, then any journey requires completing an infinite number of sub-journeys. Completing infinitely many tasks should take infinite time. Therefore, motion is impossible!
The key insight is that an infinite sum can equal a finite value. The geometric series:
The infinite sum converges to exactly 1. Similarly, the time for each sub-journey decreases proportionally—infinite steps, but executed in finite total time. Motion is rescued by calculus and limits!
Even with convergent series, the regressive version poses a deeper puzzle: if there's no first step (every step has one before it), how does motion begin? This touches on the nature of the continuum—whether space and time are truly continuous or ultimately discrete at the Planck scale.
Zeno's paradoxes forced mathematicians to develop rigorous definitions of limits, continuity, and infinite series. They remain philosophically relevant: Does the real number line capture physical space? Is time continuous? At quantum scales, perhaps space and time are granular after all—which would mean Zeno's premise of infinite divisibility is false!