THE WORLDS OF
INFINITIES

To see the world in a
grain of sand,
And a heaven in a
wildflower;
Hold infinity in the palm of you hand,
And eternity in an hour. – William Blake

Infinity has stimulated imaginations for thousands of years. It is an idea drawn upon by theologians, poets, artists, philosophers, writers, scientists, mathematicians – an idea that has perplexed and intrigued – an idea that remains illusive. Infinity has taken on different identities in different fields of thought. In early times, the idea of infinity was, rightly or wrongly, linked to large numbers. People of antiquity experienced a feeling of the infinite by gazing at stars and planets or at grains of sand on a beach. Ancient philosophers and mathematicians such as Zeno, Anaxagoras, Democritus, Aristotle, Archimedes pondered, posed and argued the ideas that infinity presented

Aristotle proposed the ideas of potential and actual infinities. He argued that only potential infinity existed.

In The Sand Reckoner Archimedes dispelled the idea that the number of grains of sand on a beach are infinite by actually determining a method for calculating the number on all the beaches of the earth.

Infinity has been the culprit in many paradoxes. Zeno’s paradoxes of Achilles and the tortoise and the Dichotomy have perplexed readers for centuries. Galileo’s paradoxes dealing with segments, points, and infinite sets should also be noted.

The list of mathematicians with their discoveries and uses or misuses of infinity extends through the centuries. (…).
Texto adaptado de PAPPAS, T. ―The Magic of Mathematics: Discovering the Spell of Mathematics‖, 1994.

Sobre as inúmeras ideias e paradoxos relativos ao infinito, o texto informa que