Exploring Zeno's Paradoxes and Foundations of Calculus

Zeno's Dichotomy Paradox

To reach a destination, you must first go halfway, then halfway again, infinitely—suggesting motion is impossible.

Achilles and the Tortoise Paradox

can never catch because he must always reach where it was, not where it is—an infinite regress.

Mathematical solution to Zeno's paradoxes

Infinite geometric series can converge to a finite value, allowing motion to be logically possible.

Countable vs. Uncountable Infinity

Countable infinity can be listed (e.g., natural numbers); uncountable cannot (e.g., real numbers).

Convergent Infinite Series

A group that adds up to a finite limit despite having infinitely many terms.

Example of a convergent series

1/2 + 1/4 + 1/8 + ... = 1

Importance of convergence in calculus

It lets infinite processes yield finite, useful results (like in limits or areas under curves).

Early calculus controversy

It used vague infinitesimals—infinitely small quantities—without rigorous grounding.

Cauchy and Weierstrass' contribution

They introduced limits to rigorously define derivatives and integrals, replacing infinitesimals.

What are infinitesimals?

Hypothetical quantities smaller than any real number but not zero, used to explain change.

Ship of Theseus

Thought experiment asking if an object remains the same when all parts are replaced.

Problem of Material Constitution

Puzzle about whether a statue and the lump of clay it's made from are the same object.

Identity vs. Persistence

is being the same object; is remaining the same over time despite change.

What is mereology?

The study of parts, wholes, and their relationships.

Mereological Universalism

The idea that any group of things, no matter how unrelated, forms a whole.

Special Composition Question

When do parts come together to form a whole?

Mereological Nihilism

The belief that only basic, partless things (simples) exist; composition never actually happens.