Why the Infinity Paradoxes Still Intrigue Us Today

Why the Infinity Paradoxes Still Intrigue Us Today

Extract of Probable Impossibilities: Thoughts on Beginnings and Endingswritten by Alan Lightman and published by Vintage Books.

The man who knows infinity

In the story of Jorge Luis Borges sand book, a mysterious stranger knocks on the narrator’s door and offers to sell him a Bible he found in a small village in India. The book shows the wear and tear of many hands. The stranger says that the illiterate peasant who gave him the book called it the Book of Sand, “because neither the sand nor this book has a beginning or an end”. Upon opening the volume, the narrator finds that its pages are creased and misplaced, with an unpredictable Arabic numeral in the upper corner of each page. The stranger suggests the narrator try to find the first page. It’s impossible.

As close to the beginning as he explores, several pages still remain between the cover and his hand. “It was like they grew from the book itself.” The stranger then asks the narrator to find the end of the book. Again, he fails. “That’s not possible,” says the narrator. “It’s not possible, but it is,” said the Bible peddler. “The number of pages in this book is literally endless. No page is the first page; no page is the last. The stranger stops and thinks. “If space is infinite, we are anywhere, anywhere in space. If time is infinite, we are at any moment. (Note to the attentive reader: we cannot be at any time. Life can only exist for a relatively short period of cosmic history, as we saw in the last chapter.)

Thoughts of infinity have hypnotized and confused human beings throughout the millennia. For mathematicians, infinity is an intellectual playground, where an infinite chain of fractions can add up to 1. For astronomers, the question is whether outer space goes on and on and on and on and on. ‘infinite. And if so, as cosmologists now believe, troubling consequences abound. For one thing, there should be an infinite number of copies of each of us somewhere in the cosmos. Because even a situation of miniscule probability – such as the creation of the exact arrangement of atoms of a particular individual – when multiplied by an infinite number of trials, repeats itself an infinite number of times. Infinity multiplied by any number (except 0) equals infinity.

Measurements at infinity are impossible, or at least impossible according to the usual notions of magnitude. If you cut infinity in half, each half is still infinite. If a weary traveler arrives at a fully occupied hotel of infinite size, no problem. You simply move the guest from room 1 to room 2, the guest from room 2 to room 3, and so on ad infinitum. In the process, you have welcomed all previous guests and released
room 1 for the newcomer. There is always room at the infinity hotel. We can play games with infinity, but we cannot visualize infinity. In contrast, we can visualize flying horses. We’ve seen horses, and we’ve seen birds, so we can mentally implant wings on a horse and send it flying. This is not the case with infinity. The invisibility of infinity is part of its mystique.

The first recorded conception of infinity appears to have taken place around 600 BC and is attributed to the Greek philosopher Anaximander, who used the word apeiron, meaning “unlimited” or “unlimited”. For Anaximander, the Earth and the heavens and all material things were caused by infinity, although infinity itself is not a material substance. Other ancient Greek philosophers held that infinity was a negative, even an evil, because the inability to measure something was considered a defect of the thing – except for the infinite and immeasurable One. Around the same time as
Anaximander, the Chinese used the word wuji, meaning “unlimited”, and wuqiong, meaning “endless”, and believed that infinity was very close to nothingness. (An interesting perspective on Pascal’s ideas, discussed in “Between Nothingness and Infinity.”) In Chinese thought, being and non-being, like yin and yang, are in harmony one with the other – hence the kinship of infinity and nothingness.

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A few centuries later, Aristotle argued that infinity does not really exist. He conceded something he called potential infinity, like whole numbers. For any number, you can always make a larger number by adding one to it. This process can continue as long as your stamina holds, but you can never reach infinity. Indeed, one of the many intriguing properties of infinity is that you can’t get there from here. The infinite is not simply more and more of the finite. It appears to be of a completely different nature, although pieces of it may appear finite, like large numbers, or like large volumes of space. Infinity is a thing in itself. Everything we see and experience has limits, limits, tangibilities. This is not the case with infinity. For similar reasons, Saint Augustine, Spinoza and other theological thinkers associated the infinite with God: the unlimited power of God, the unlimited knowledge of God, the unlimited of God. “God is everywhere and in all things, insofar as he is unlimited and infinite”, said Saint Thomas Aquinas.

Beyond the religious sphere of the immaterial world, physicists believe that there can also be infinite things in the material world. But this belief can never be proven. You cannot get there from here. Most of us have our first glimmers of infinity when we are children, when we look at the night sky for the first time. Or when we go out to sea, out of sight of land, and watch the ocean stretch endlessly until it meets the horizon. But these are only glimmers, like counting to a few thousand in Aristotle’s infinite potential. We are overwhelmed. But we did not approach.

The concept of infinity remains a controversial and paradoxical subject today, galvanizing international conferences and fiery scholarly disputes. Can physical forces ever be infinite in strength? Can physical space expand beyond boundless galaxy after galaxy? Is there an infinity between the infinity of whole numbers and the infinity of all numbers? In May 2013, a panel of scientists and mathematicians met in New York to discuss the deep enigmas surrounding infinity. William Hugh Woodin, a mathematician at the University of California at Berkeley, put it this way: “It’s a bit like mathematics living on a stable island: we’ve built a solid foundation. Then there is the wild land there. It’s infinity.

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