In Folha, Viana discusses the paradoxes of mathematics.

Foto: Agência Brasil

Reproduction of Marcelo Viana's column in Folha de S.Paulo.

The swift Achilles races the tortoise. The tortoise starts 100 meters ahead, but Achilles covers that distance in 10 seconds. When he reaches the point where the tortoise started, the tortoise has already moved 1 meter. Achilles covers this distance in 0.1 seconds, but again, when he arrives where the tortoise was, it is no longer there. Thus, despite being faster, Achilles never catches the tortoise.

This and other famous paradoxes, attributed to the Greek philosopher Zeno of Elea (490-430 BC), aimed to show that change, and in particular movement, are illusory. This is in direct opposition to the thought of contemporaries such as Heraclitus of Ephesus (500-450 BC), who inspired the aphorism "everything flows like a river".

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Most mathematicians believe that these paradoxes were resolved with the discovery of calculus, but the philosophical discussion continues to this day. And recent advances in quantum mechanics point in directions strangely reminiscent of Zeno's ideas.

In Christian tradition, the Last Judgment will be announced by the sounding of the trumpet of the archangel Gabriel. This is undoubtedly a remarkable instrument, probably infinite. The Italian mathematician Evangelista Torricelli (1608-1647) considered it to be the solid obtained when the graph of y=1/x is rotated around the x-axis. In this case, Gabriel's trumpet would have finite volume (it holds a finite amount of air), but infinite area (painting it requires an infinite amount of paint). This conclusion gave rise to a controversy with various thinkers of the time, including Galileo Galilei (1564-1642), about the meaning of infinity.

One day, in college, my professor José Morgado (1921 – 2003) proved the following theorem: "if 3=2 then I am the Pope".

The idea is this: if 3=2 then, subtracting one, 2=1; therefore, the Pope and I, who are two people, are actually one. The moral is that from false assumptions it is possible to arrive at any conclusion, using the most perfect logic. This lesson applies to life.

An ant walks at a speed of 1 cm/s inside a rubber tube that is initially 1 km long. To make it more difficult, the tube is stretched at a rate of 1 km/s, meaning that every second it gets 1 km longer. In this way, the ant will never reach the end of the tube, right?

Wrong: it takes a long time, but it does arrive! Can the reader explain this?