# Zeno and Melissus

Philosophy I · Ancient Philosophy: From the Milesians to Plato · https://tryals.app/learn/philosophy-i/zeno-and-melissus

## Defending the Indefensible by Attacking the Obvious

Zeno of Elea was Parmenides' student, and his paradoxes are usually misunderstood. He was not trying to convince anyone that motion is impossible. He was arguing **dialectically**: his opponents said Parmenides' position was absurd, so Zeno showed that *their* position — that there are many things, and that they move — leads to absurdities just as bad. If both views are paradoxical, ridiculing Parmenides is no longer an argument.

The **Dichotomy**: to reach the end of a path you must first reach the halfway point, then half the remainder, and so on without end. Completing infinitely many tasks in finite time seems impossible, so motion cannot begin.

**Achilles and the Tortoise**: give the tortoise a head start. By the time Achilles reaches where it was, it has moved a little further. Repeat forever. Achilles never passes it.

The **Arrow**: at any instant, a flying arrow occupies a space exactly its own size, and so is at rest. Time is composed of instants. So the arrow is at rest at every instant of its flight.

The modern resolution of the first two is that an **infinite series can have a finite sum**:

$$\tfrac{1}{2} + \tfrac{1}{4} + \tfrac{1}{8} + \dots = 1$$

Infinitely many terms, finite total. The intervals shrink fast enough that the whole journey takes finite time. This is the geometric series $\sum 2^{-n}$, and it converges, a fact Zeno had no way to state.

Whether that fully answers him is still argued. It shows the *time* is finite; whether it explains how infinitely many tasks are *completed* is a separate question, and the Arrow — which is about what motion consists in at an instant, not about summation — resists the treatment entirely.

**Melissus** defended the same position by different means, arguing that what-is must be **spatially infinite** (since a limit would be a boundary with what-is-not) and therefore incorporeal, since a body would have parts. He also gave the argument that if things really changed, then what-is would become what-is-not, so the very possibility of change refutes itself.

> **Common pitfall:** thinking Zeno was refuted the moment convergent series were understood. The mathematics settles that the *sum of the times* is finite. It does not obviously explain how a supertask, infinitely many distinct completions, is performed, and the Arrow paradox is untouched by it.

## Practice questions

7 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. Why did Zeno construct the paradoxes, given that he did not expect anyone to stop believing in motion?

A. To demonstrate that sensory perception is far more reliable than abstract reason
B. To show that his opponents’ position is as paradoxical as the one they ridiculed
C. To establish a definitive mathematical proof that physical motion is strictly impossible
D. To show that geometry cannot coherently describe the continuous nature of physical space

**Answer:** B. To show that his opponents’ position is as paradoxical as the one they ridiculed

**Why:** The argument is dialectical: if plurality and motion generate paradoxes just as severe, then ridiculing Parmenides for absurdity is no longer an argument. Zeno is levelling the field, not proving a positive thesis.

Page: https://tryals.app/practice/philosophy-i/zeno-and-melissus/why-did-zeno-construct-the-paradoxes-given-that-he-did-not-expect

### 2. Achilles runs at 10 metres per second and the tortoise at 1 metre per second with a 100 metre head start. After how many seconds does Achilles draw level?

**Answer:** 11.11 (within ±0.2)

**Why:** The closing speed is $10 - 1 = 9$ m/s over a 100 m gap, so $100/9 = 11.11$ s. The catch-up happens at a perfectly definite moment. Zeno’s infinite subdivision describes only the time *before* it.

Page: https://tryals.app/practice/philosophy-i/zeno-and-melissus/achilles-runs-at-10-metres-per-second-and-the-tortoise-at-1-metre-per

### 3. A convergent series demonstrates that an infinite sequence of shrinking intervals yields a finite total. What follows from this mathematical result for Zeno's project?

A. It disproves the Dichotomy by showing that spatial division must be impossible
B. It fully refutes the Arrow paradox by dissolving the need for indivisible points
C. It settles the duration of the journey whilst leaving the supertask unresolved
D. It proves that Eleatic monism is physically coherent through formal summation

**Answer:** C. It settles the duration of the journey whilst leaving the supertask unresolved

**Why:** Convergent series account for total duration, but modern philosophers still dispute whether performing infinitely many distinct sub-tasks is coherent. Furthermore, the Arrow targets the nature of motion at an instant, an issue untouched by series summation.

Page: https://tryals.app/practice/philosophy-i/zeno-and-melissus/a-convergent-series-demonstrates-that-an-infinite-sequence-of

### 4. Which claims about the modern treatment of Zeno’s paradoxes are correct?

A. The paradoxes were shown to rest on arithmetic errors
B. The Arrow paradox is not resolved by summation
C. This addresses the Dichotomy and the Achilles
D. The infinite series of times converges to a finite total

**Answer:** B. The Arrow paradox is not resolved by summation; C. This addresses the Dichotomy and the Achilles; D. The infinite series of times converges to a finite total

**Why:** Convergence handles the two subdivision paradoxes and leaves the Arrow untouched, since the Arrow concerns what motion *is* at an instant rather than how intervals sum. Zeno made no arithmetic error at all.

Page: https://tryals.app/practice/philosophy-i/zeno-and-melissus/which-claims-about-the-modern-treatment-of-zenos-paradoxes-are

### 5. Melissus agreed with Parmenides that what-is is complete and bounded, like a well-rounded sphere.

**Answer:** False

**Why:** False. Melissus departs from Parmenides here, arguing that what-is must be spatially **infinite**. A limit requires something beyond it to be limited *by*, and on Eleatic principles that could only be what-is-not, which is forbidden.

Page: https://tryals.app/practice/philosophy-i/zeno-and-melissus/melissus-agreed-with-parmenides-that-what-is-is-complete-and-bounded

### 6. Arrange the steps of the Achilles paradox in order.

**Answer:**

1. The tortoise is given a head start
2. Achilles runs to where the tortoise began
3. By then the tortoise has moved a little further on
4. Achilles runs to that new position, and the tortoise moves again
5. The sequence never terminates, so Achilles never passes it

**Why:** The trick is that each stage ends where the tortoise *was*, so by construction the sequence never reaches the overtaking. The stages describe only the time before the catch-up, which arrives on schedule regardless.

Page: https://tryals.app/practice/philosophy-i/zeno-and-melissus/arrange-the-steps-of-the-achilles-paradox-in-order

### 7. Sort each statement by whether it accurately describes Zeno’s paradoxes.

**Answer:**

- Accurate: They were intended to defend Parmenides dialectically, They target the assumptions of plurality and motion, The Arrow concerns what motion consists in at an instant
- Inaccurate: Zeno believed he had proved that nothing ever moves, They rest on an arithmetical mistake about infinite series

**Why:** Zeno’s aim was defensive and his reasoning contained no error, which is what makes the paradoxes durable. They are still discussed because the questions about supertasks and the composition of time were never fully closed.

Page: https://tryals.app/practice/philosophy-i/zeno-and-melissus/sort-each-statement-by-whether-it-accurately-describes-zenos