Deep, and method-defining. Zeno gave the first systematic proofs by contradiction (Aristotle calls him the inventor of dialectic), and the first serious confrontation with the infinite — problems that drove the eventual rigor of limits and the calculus.
Answered by limits. The convergent-series resolution (½+¼+…=1) handles Achilles and the dichotomy cleanly. But it took Cauchy and Weierstrass to make it rigorous, and the arrow and "supertasks" still feed live debate in philosophy.
Polemics, in service of Parmenides. Zeno's paradoxes argue against motion/plurality to protect his teacher's One; they're brilliant reductios, not a positive theory. And we have them only second-hand, mostly via Aristotle.
The tortoise gets a head start; Achilles runs ten times faster. By the time he reaches the tortoise's start, it has moved a little; by the time he reaches that spot, it has moved again — forever. Zeno concludes he never catches it. Press run and watch the stages: each is real, there are infinitely many — yet their lengths shrink so fast that their total is finite, and Achilles passes the tortoise at a definite point.
"That which is in locomotion must arrive at the half-way stage before it arrives at the goal." — Zeno's Dichotomy, via Aristotle
To go anywhere you must first go halfway, but first half of that, and so on — infinitely many sub-journeys to complete before you even start. So motion can't begin.
The swift can never catch the slow with a head start: each time he reaches its last position, it has moved on. Infinitely many catch-ups to make.
At any single instant a flying arrow occupies exactly its own length and is at rest. If it's at rest at every instant, when does it move? Motion vanishes into a sum of still frames.
All three attack the same target: the idea that a continuous quantity (a distance, a duration) is built of infinitely many parts. The dichotomy and Achilles are answered by convergent series — infinitely many shrinking steps with a finite sum. The arrow is subtler: it forces a real account of what "motion at an instant" even means, which is precisely the derivative — instantaneous velocity, defined as a limit. Twenty-three centuries after Zeno, the calculus didn't just refute him; it gave back rigorous answers to the exact questions he was the first to ask. ⚑ A "wrong" argument so good it had to invent a branch of mathematics to be properly answered — the highest compliment a paradox can earn.
Gate kept on. The honest reading gives Zeno enormous credit while marking the limits. His method is foundational: Aristotle himself called Zeno the inventor of dialectic, and the paradoxes are among the first rigorous proofs by contradiction — assume motion/plurality, derive an absurdity. His confrontation with infinity was 2,000 years ahead of the tools needed to handle it. The honest qualifications: first, the paradoxes are not unsolved — for the dichotomy and Achilles, the convergent geometric series (an infinite sum of shrinking terms with a finite total) is a complete and correct resolution, made fully rigorous by Cauchy and Weierstrass in the 1800s; pop claims that "math can't really answer Zeno" are wrong. Second, they are partly alive: the arrow and the broader puzzle of supertasks (completing infinitely many actions in finite time) remain genuinely debated in philosophy of mathematics, so it's not all tidily closed. And third, like his teacher, Zeno's aim was defensive — these are weapons forged to protect [[parmenides]]' unchanging One, not a positive theory of their own, and we read them mostly through Aristotle's reports. Strip all that away and the achievement stands: the man who first made infinity a problem, and whose "impossible" arguments were so sharp they helped summon the calculus into being. ⚑ With Zeno the front of the lineage is whole; the road runs on to its first systematiser. [[parmenides]] ← · next → [[aristotle]] (the science of valid inference).