Contemplating Infinity
 By Peter Tyson
 Posted 09.30.03
 NOVA
How can we get our heads around the mindbending notion of the infinite?
If in the infinite you want to stride,
Just walk in the finite to every side.
—Goethe
The word after "infinity" in my dictionary is "infirm," a definition of which is "weak of mind." This is how many of us who are not mathematically inclined feel upon contemplating infinity.
Quita Brodhead. The Endless Circle Series #5. 1987. Oil on canvas. 501/2 by 281/2 inches. Enlarge Photo credit: Courtesy of Hollis Taggart Galleries, New York, and the Estate of Quita Brodhead
We feel weak because our minds can only go so far with the conjecture, and because every time we think we're on the verge of securing even a shadowy understanding, we're tripped up by something. A friend of mine once told me that trying to hold her hyperactive toddler was like trying to hold a live salmon. Infinity is like that for us "infirm ones": slippery as a salmon, forever eluding our grasp.
Becoming numb
This is true no matter how you approach the concept. Many of us might consider numbers the most surefooted way to come within sight of infinity, even if the mathematical notion of infinity is something we'll never even remotely comprehend.
We may think, for starters, that we're well on our way to getting a sense of infinity with the notion of no biggest number. There's always an ever larger number, right? Well, no and yes. Mathematicians tell us that any infinite set—anything with an infinite number of things in it—is defined as something that we can add to without increasing its size. The same holds true for subtraction, multiplication, or division. Infinity minus 25 is still infinity; infinity times infinity is—you got it—infinity. And yet, there is always an even larger number: Infinity plus 1 is not larger than infinity, but 2^{infinity} is.
Try another tack: huge numbers. When we play with mindboggling figures we nonmath types may think we're playing in infinity's neighborhood, if not in the same playground. When we're told that the decimals in certain significant numbers, like pi and the square root of two, go on forever, we can somehow accept that, especially when we learn that computers have calculated the value of pi, for one, to over a trillion places, with no final value for pi in sight. When we're told that there are 43,252,003,274,489,856,000 possible ways to arrange the squares on the Rubik Cube's six sides, we may feel intuitively (if not rationally) that we must be on our way to the base of that loftiest of all peaks, Mt. Infinity.
Quita Brodhead. Ascension. 1994. Oil on canvas. 36 x 48 inches. Enlarge Photo credit: Courtesy of Hollis Taggart Galleries, New York, and the Estate of Quita Brodhead
One reason we may feel this way is that such numbers are as intellectually unapproachable to the mathematically challenged as infinity itself. Take a Googol. A Googol is 10^{100}, or 1 followed by 100 zeroes, and is the largest named number in the West. The Buddhists have an even more robust number, 10^{140}, which they know as asankhyeya. Just for fun, I'll name a larger number yet, 10^{1000}. I'll call it the "Olivian," after my daughter. Now, doesn't an Olivian get me a little closer to infinity than the Googolians or even the Buddhists can get? Nope. Infinity is just as far from an Olivian as it is from a Googol—or, for that matter, from 1.
Many of us are as queasy around eternity as we are around infinity.
Perhaps we infirm ones would be wise to take a leaf from the lingual book of Madagascar. The word there for a million is tapitrisa, which means "the finishing of counting." For some tribal groups in other parts of the world, counting stops at three, in fact; anything above that is "many." In some ways this makes sense. How many of us can keep more than a few things in our minds at once? I remember playing a game with myself as a child in which I would think "I'm thinking that I'm thinking that I'm thinking that I'm thinking...." After the third or fourth "I'm thinking," I could no longer retain in my head all the degrees it implies. Such infirmity holds for simple counting as well, as Lewis Carroll reveals so tellingly in Through the Looking Glass:
"Can you do Addition?" the White Queen asks. "What's one and one and one and one and one and one and one and one and one and one?"
"I don't know," said Alice. "I lost count."
"She can't do Addition," the Red Queen interrupted. "Can you do Subtraction?"
For many of us uncomfortable with infinity, the word number can be defined as "that which makes numb," as Rudy Rucker wryly notes in his book Infinity and the Mind. This is especially true when a number is so outlandishly enormous that it smacks, however remotely, of the infinite. Galileo himself felt this way. "Infinities and indivisibles transcend our finite understanding, the former on account of their magnitude, the latter because of their smallness," he wrote in his Dialogues of Two New Sciences of 1638. "Imagine what they are when combined." Rather not, thanks—makes me numb.
Quita Brodhead. The Endless Circle Series #4. 1987. Oil on canvas. 48 x 36 inches. Enlarge Photo credit: Courtesy of Hollis Taggart Galleries, New York, and the Estate of Quita Brodhead
Incredible shrinking
Infinities do come in two sizes, of course—not only the infinitely large but also the infinitely small. As Jonathan Swift wrote, "So, naturalists observe, a flea/Has smaller fleas that on him prey/And these have smaller still to bite 'em/And so proceed ad infinitum." We may not be able to conceive of Swift's infinitesimal fleas, because reason insists they don't exist, but we can imagine ever smaller numbers without much trouble. It's no hardship, for example, to grasp the notion of an infinity of numbers stretching between, say, the numerals 2 and 3. Take half of the 1 that separates them, we might tell ourselves, then half of that half, then half of that half, and so proceed ad infinitum.
Of course, just when we think we have infinity in the palm of our hands, we watch it evaporate in the harsh light of another of those confounding paradoxes: The numerals 2 and 3 are separated by both a finite number—1—and an infinity of numbers. This conundrum spawned one of the great Gordian knots in early philosophical history, known as Zeno's paradox. Zeno was a Greek philosopher of the fourth century B.C. who "proved" that motion was impossible. For a runner to move from one point to another, Zeno asserted, he must first cover half the distance, then half the remaining distance, then half the remaining distance again, and so on and so on. Since this would require an infinite number of strides, he could never reach his destination, even if it lay just a few steps away.
It wasn't for 2,000 years that Zeno's paradox finally got "solved," for all intents and purposes, by the calculus. Its independent inventors, Isaac Newton and Gottfried Leibniz, showed us how an infinite sum can add up to a finite amount, that it can converge to a limit. Thus, even though we can't count all the numbers between 2 and 3, we know they converge to 1.
No limits
As Zeno's paradox hints, considering infinity from the perspective of space has much correspondence with that of numbers. We can imagine, for instance, that space, like numbers, is infinitely divisible. We understand Hamlet when he says "I could be bounded in a nutshell/And count myself a king of infinite space." The shortest length physicists speak of is the Planck length, 10^{33} centimeters. But might not there be an even shorter length, say, 10^{333} centimeters, or 10^{an infinite number of 3's} centimeters?
Quita Brodhead. Whence and Where To. 2000. Oil on canvas. 361/4 x 48 inches. Enlarge Photo credit: Courtesy of Hollis Taggart Galleries, New York, and the Estate of Quita Brodhead
As with numbers, we can also envision space as being infinitely large. That's somehow easier to accept than that space ends at some sort of boundary, and that there's no space on the other side. If not space, then what? We also might flatter ourselves that we're somehow getting closer to infinity when we consider extremely large distances. On June 12, 1983, while traveling at over 30,000 mph, the Pioneer 10 spacecraft became the first humanmade object to exit our solar system. Some 300,000 years from now, unless something interrupts its voyage, the craft is expected to pass near the star Ross 248, a red dwarf in the constellation Taurus. Ross 248 is about 10.1 lightyears from Earth, or about 59,278,920,000,000 miles away. Pioneer 10 will still be in the early stages of its journey, though. When our sun bloats into a red giant about five billion years from now and incinerates our planet, our robotic ambassador will still be heading away, knocking off more than 250 million miles a year.
"Infinity is where things happen that don’t."
Are we making headway towards an infinite distance with such knowledge? Hardly. An infinite distance, as you've guessed, would be as far from where Pioneer 10 will be in five billion years as it is from the Earth now. If the universe is infinitely large, even the remotest stars we can detect, which are so far away that their light left them some 12 billion years ago, are as far from infinity as we are. (Things get tricky here: As one mathematician pointed out to me, infinity is an abstract concept, appearing only in our mental images of the universe. It is not actually in the universe.)
Forever and a day
Time is another way to contemplate infinity, though many of us are as queasy around eternity as we are around infinity. ("That's the trouble with eternity, there's no telling when it will end," Tom Stoppard writes in Rosencranz and Guildenstern Are Dead.) Yet isn't infinite time somehow easier to swallow than finite time? After all, what can stop time?
Many of us do indeed live our lives thinking that eternity is a given. And again, we may fool ourselves into thinking that we're on the way to eternity when we think of 12 billion years, or of any other frighteningly mindbending length of time. One of the gamest attempts to define eternity I've seen appears in Hendrik Willem Van Loon's 1921 children's classic The Story of Mankind:
High up in the North in the land called Svithjod, there stands a rock. It is 100 miles high and 100 miles wide. Once every thousand years a little bird comes to the rock to sharpen its beak. When the rock has thus been worn away, then a single day of eternity will have gone by.
That passage gives you an inkling for just how goshdarn long eternity is. But all the usual caveats apply: Eternity doesn't have a length, that single "day" of eternity is as far in time from eternity itself as a normal day, and so on.
No fear of the infinite: Quita Brodhead was an abstract painter who investigated infinity in her "Endless Circle" series and other paintings, several of which illustrate this article. Brodhead died in September 2002 at the age of 101. Enlarge Photo credit: © Peggy Fox
Fear of the infinite
If all this leaves you feeling numb, you're not alone. The Greeks, in fact, invented apeirophobia, fear of the infinite. (The term comes from the Greek word for infinity, apeiron, which means "without boundary.") Aristotle would only admit that the natural numbers (1, 2, 28, etc.) could be potentially infinite, because they have no greatest member. But they could not be actually infinite, because no one, he believed, could imagine the entire set of natural numbers as a finished thing. The Romans felt just as uncomfortable, with the emperor Marcus Aurelius dismissing infinity as "a fathomless gulf, into which all things vanish."
"Infinity is a floorless room without walls or ceiling."
The ancients' horror infiniti held sway through the Renaissance and right up to modern times. In 1600, the Inquisitors in Italy deemed the concept so heretical that when the philosopher Giordano Bruno insisted on promulgating his thoughts on infinity, they burned him at the stake for it. Later that century, the French mathematician Blaise Pascal deemed the concept truly disturbing: "When I consider the small span of my life absorbed in the eternity of all time, or the small part of space which I can touch or see engulfed by the infinite immensity of spaces that I know not and that know me not, I am frightened and astonished to see myself here instead of there ... now instead of then." Martin Buber, an Israeli philosopher who died in 1965, felt so undone by the concept of infinity that he "seriously thought of avoiding it by suicide."
Most of us will never feel so put out by infinity that we'll resort to contemplating such extreme measures. We may feel weak of mind, like the anonymous schoolboy who once declared that "infinity is where things happen that don't." But our uneasiness will never get much greater than the schoolboy's delightfully dismissive attitude suggests his got. We can live with that level of discomfort, contenting ourselves with the knowledge that all we can reasonably expect in musing on infinity is to get a feeling for it, like that engendered by this gem from another anonymous sufferer of our common infirmity: "Infinity is a floorless room without walls or ceiling."
This feature originally appeared on the site for the NOVA program Infinite Secrets.
Sources
Broyard, Anatole. 1987. "Infinity is bigger than all of us." The New York Times. 19 April 1987.
Maor, Eli. 1987. To Infinity and Beyond: A Cultural History of the Infinite. Birkhäuser.
Moore, A. W. 1995. "A brief history of infinity." Scientific American. Vol. 272, No. 4 (April 1995), 112116.
Pickover, Clifford A. 1995. Keys To Infinity. John Wiley & Sons.
Rotman, Brian. 1997. "The truth about counting." The Sciences. Vol. 37 No. 6 (NovemberDecember 1997), 3439.
Rucker, Rudy. 1982. Infinity and the Mind: The Science and Philosophy of the Infinite. Birkhäuser.
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