Why the infinity paradoxes still baffle us today – Big Think

Adapted from Possible Impossibles: Reflections on Beginnings and EndsWritten 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 the Bible he brought in a small village in India. The book shows the erosion of the hands. The stranger says that the illiterate peasant who gave him the book called it the Book of Sand, “because neither sand nor this book have a beginning nor an end.” Upon opening the volume, the narrator finds that its pages are wrinkled and poorly defined, with an unexpected Arabic numeral in the top corner of each page. The stranger suggests that the narrator is trying to find the first page. No way.

No matter how close to the beginning he explores, there are always several pages left between the cover and his hand. “It was as if they had both originated from the same book.” Then the stranger asks the narrator to find the end of the book. Once again, it failed. “It can’t be,” says the narrator. It can’t be, says the Bible peddler, “but it is.” “The number of pages in this book is literally unlimited. No page is the first page; no page is the last.” The stranger stops and thinks. “If space is infinite, we are anywhere and at any point in space. If time is infinite, we are at any time.” (Note to the attentive reader: We can never be. Life can only exist during a relatively short period of cosmic history, as discussed in the last chapter.)

Ideas of infinity have fascinated and confused humans for thousands of years. For mathematicians, infinity is a mental playground, where an endless series of fractions can add up to 1. For astronomers, the question is whether outer space goes on and on and on and on to infinity. And if that happens, as cosmologists now believe, the worrying consequences will be many. First, there must be an infinite number of copies of each of us somewhere in the universe. Because even the case of tiny probability – such as the formation of the exact order of the atoms of a given individual – when multiplied by an infinite number of trials, it repeats itself an infinite number of times. Multiplying infinity by any number (except 0) equals infinity.

Measurements of infinity are impossible, or at least impossible according to the usual concepts of volume. If you cut infinity in half, then each half will remain infinite. If a weary traveler arrives at a fully occupied hotel of infinite size, no problem. You can simply move the guest in room 1 to room 2, the guest in room 2 to room 3 and so on indefinitely. In the process, you absorb all the previous guests and free yourself
Room 1 for new arrivals. There is always a room in the infinity hotel. We can play games infinity, but we cannot imagine infinity. By contrast, we can imagine flying horses. We’ve seen horses and we’ve seen birds, so we can mentally plant wings on a horse and send it high. Not so with infinity. Part of its ambiguity is the lack of perception of infinity.

The first recorded conception of infinity appears to have occurred around 600 BC and is attributed to the Greek philosopher Anaximander, who used the word apeiron, meaning “unlimited” or “unlimited”. For Anaximander, the earth, the heavens, and all material things were their cause of infinity, even though infinity itself was not a material substance. Other Greek philosophers considered infinity to be a negative, rather an evil, because the inability to measure something was considered a defect in the thing—except for the immeasurable infinite. around the same time
Anaximander, the Chinese used wuji, which means “without limits”, and wuqiong, which means “endless”, and they believed that the infinite is very close to nothingness. (An interesting perspective on Pascal’s ideas, discussed in “Between Nothingness and Infinity.”) In Chinese thought, being and nothingness, like yin and yang, are in harmony with each other—hence infinite kinship and nothingness.

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Several centuries later, Aristotle argued that infinity does not actually exist. He admitted something he called potential infinity, like integers. For any number, you can always create a larger number by adding one to it. This process can go on as long as your stamina holds, but you can never go to infinity. In fact, among the many interesting properties of infinity is that you cannot get there from here. Infinity is not just more and more limits. It appears to be of an entirely different nature, although parts of it may appear finite, like large numbers, or like large expanses of space. Infinity is a thing in itself. Everything we see and experience has limits, boundaries, and tangibles. Not so with infinity. For similar reasons, St. Augustine, Spinoza, and other theological thinkers have linked infinity to God: the infinite power of God, the infinite knowledge of God, and the infinity of God. Saint Thomas Aquinas said: “God is omnipresent and in everything, inasmuch as he is infinite and infinite.”

Outside of the religious realm of the non-physical world, physicists believe that there may be infinite things in the physical world as well. But this belief can never be proven. You cannot get there from here. Most of us have our first glimpses of infinity as children, when we look up at the night sky for the first time. Or when we go out to sea, out of sight of land, and stare at the ocean that stretches incessantly until it meets the horizon. But these are only glimpses, like counting to a few thousand at Aristotle’s potential infinity. We are tired. But we did not come close.

The concept of infinity remains a controversial and contradictory topic today, resulting in international conferences and heated scientific controversies. Can bodily forces be limitless in strength? Could physical space extend beyond galaxy after galaxy infinitely? Is there an infinity between the infinity of integers and the infinity of all numbers? In May 2013, a panel of scientists and mathematicians met in New York City to discuss the deep mysteries surrounding infinity. William Hugh Woden, a mathematician at the University of California, Berkeley, put it as follows: “It’s kind of like mathematicians live on a stable island — we’ve built a solid foundation. Then there’s the wild land outside. That’s infinity.”

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