Contemplating the Mathematics of Infinity – The Story of Zeno

Infinity in Ancient Mathematics

What is infinity? What does it mean to say that something is without limit, boundless and endless in space and time? 

Mankind has tried to answer this in many ways through philosophy and mathematics. The ancient Indian Vedic philosophers and the ancient Greek mathematicians were the two groups who made the most innovative discoveries about infinity. However, their conclusions were radically different.

For most ancient Greeks, infinity was a scary concept. It led, they believed, to chaos. The ancient Greeks believed in an ideal of harmonious objects that could be clearly understood using the counting numbers, 1,2,3,…. or fractions of counting numbers. Legend has it that when one of the Pythagorean mathematicians discovered a number (the square root of two) that could not be written as a fraction of counting numbers and required an infinite series to describe it, he was drowned. The Pythagoreans tried to keep the number’s existence secret.

For the Vedic philosophers, infinity was a form of divinity. The infinite transcends birth and death, beginnings and ends, and is the true reality. The scholar G. S. Pandey writes that “A number of abstract ideas like the origin and shape of the universe, meaning of time, [and infinity] were the subject matter of academic discussions during the Vedic period. It is universally accepted that the concept of infinity, zero, and decimal system are the original contributions of ancient Indian mathematicians, which have played a very vital role in the progress of human civilization and culture on this planet.”

Zeno’s Paradox

Zeno of Elea (495 BCE—430 BCE) was a Greek philosopher and mathematician. An ancient biographer, Diogenes Laertius, wrote about Zeno that he was “skilled to argue both sides of any question.” Laertius said that Zeno was part of a group that tried to overthrow a local tyrant. Zeno was arrested and was being tortured into giving up the names of his fellow conspirators. Zeno refused but said to the tyrant that he had another secret to reveal. When the tyrant came close to Zeno to hear this secret, Zeno is said to have bitten the tyrant’s ear! 

Zeno created around forty paradoxes, of which only ten are still known. The most famous paradox that involves infinities is as follows:

Achilles, the great Greek warrior, and a tortoise agree to a race. Since Achillies is so fast, the tortoise is given a lead of 100 meters. When the race starts, Achilles quickly reaches the 100-meter mark, but in that time, the tortoise has moved forward 50 meters. Achilles then rapidly reached the 150-meter mark, but in that time, the tortoise has gone forward 25 meters. Every time Achilles reaches where the tortoise was; the tortoise has gone forward a small distance ahead. 

In this way, Zeno’s paradox states that even the fastest of runners can never over­take the slowest of runners because the fast runner must first reach the point where the slow runner was, so the slow runner will always be ahead. 

Of course, we know that a fast runner easily catches up, over-takes, and goes ahead of a slow runner. So, where is the flaw in the argument? The solution of the paradox hinges on an understanding of inifinity and leads to one of modern mathematics’s greatest inventions – calculus. 

Calculus and the Taming of Infinities

The “solution” to Zeno’s paradox is that even an infinite series of additions say the series 1/2 + 1/4 + 1/8 + 1/16 … is equal to a finite number, in this case, 1. Achilles reduces the distance between himself and the tortoise at a faster and faster rate. Thus these infinite additions end up being equal to 1.

The invention of calculus that overcomes Zeno’s paradox is one of the highest achievements of human beings. While parts of calculus were known for thousands of years, it was Isaac Newton and Gottfried Leibniz who in the last 1600s made the most significant advances. It is due to the applications of calculus that our science and technology were able to advance and for us to have the wonders of the modern world.

Other Advances   

Mathematicians built on the foundations of Newton and Leibniz and there are many fascinating discoveries. Two of the most astonishing are:

  • Georg Cantor (1845-1918) was interested in the infinite from a young age. He discovered that there are different types of infinities and that some infinities are larger than others. 
  • Quantum Computers use the latest technology and the physics of the very small (quantum mechanics) to develop a new type of computer. Such computers are able to solve certain types of problems in a fraction of the time it takes conventional computers. Quantum computers can, theoretically, solve problems involving infinite dimensions.

Unfortunately, most of the above will be difficult for readers to understand deeply. In school, mathematics is taught terribly, and most of us then leave the subject aside. Yet mathematics reveals the true beauty of the infinite.

For example, the brilliant Indian mathematician Srinivasa Ramanujan (1887-1920) proved that if you add all the numbers 1, 2, 3, 4,.. etc. all the way to infinity, the answer is equal to: -1/12.  This is now called the Ramanujan Summation and it plays an important role in areas of modern physics.

Infinity in the palm of your hand

Perhaps it is best to end this week’s article with the opening lines of William Blake’s poem, “Auguries of Innocence.” The poem captures the process of how science and mathematics contemplate and reveal infinity:

“To see a World in a Grain of Sand

And a Heaven in a Wild Flower 

Hold Infinity in the palm of your hand 

And Eternity in an hour” 

© Kaikhushru Taraporevala

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