How Big Is infinity?
Becoming numb
This is true no matter how you approach the concept. Many of us might consider numbers the most sure-footed 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 2infinity is.
Try another tack: huge numbers. When we play with mind-boggling figures we non-math 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.
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 10100, or 1 followed by 100 zeroes, and is the largest named number in the West. The Buddhists have an even more robust number, 10140, which they know as asankhyeya. Just for fun, I'll name a larger number yet, 101000. 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.
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 (Birkhäuser, 1982). 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.
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 paradoxes of 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 strides away.
It wasn't for 2,000 years that Zeno's paradox finally got "solved," for all intents and purposes, by the calculus. Its 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.
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 believe 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?
As with numbers, we can also envision space as being infinitely large. After all, if the universe has a boundary, what's on the other side? We 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 human-made 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 light-years 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 its own red dwarf 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.
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.)
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 mind-bending length of time. One of the gamest attempts to define eternity 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 gosh-darn 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, etc., etc.
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."
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."
For mathematicians, infinity means something completely different than for philosophers. It's not something vague and unapproachable, but rather something with a precise definition that lies at the core of modern mathematics. To explain how this eminently practical form of infinity evolved over time, and to translate for the layman how mathematicians think of it, we approached Stanford University classics historian Reviel Netz. A scholar who discovered that, contrary to belief, the ancient Greeks, through the work of Archimedes, had actually toyed with infinitely large sets, Netz knows a thing or two about mathematical infinity..
n: How do mathematicians define infinity?
m: Something which is equal to some of its parts. That's really the technical definition.
n: Is there a difference between the mathematical concept of infinity and infinity in terms, say, of space or time—the philosophical concept?
m: That's the curious thing. Infinity became a really clear and well-defined quantity mathematically in the late 19th century, which it wasn't before and which makes it rather different from what we ordinarily talk about when we talk about infinity, namely, about something very, very big. In mathematics nowadays, when we think about infinity, we think about a set whose properties are different from those of ordinary sets.
n: Can you explain?
m: Well, the defining property of infinity today is that a set's cardinality [the number of elements in a given mathematical set] is equal to the cardinality of some real subset of that set. The thread that links the technical notion of infinity and the more popular/philosophical notion of infinity is the sense that infinity's hugeness being beyond reach endows it with certain paradoxical properties.
A standard example would be what was taken to be something rather paradoxical, the fact that you can take, say, the number 1 and correlate it with the number 2, take the number 2 and correlate it with the number 4, take the number 3 and correlate it with the number 6. In this way you can gradually, step by step, correlate all integer numbers (1, 2, 3, 4, etc.) with all even numbers (2, 4, 6, 8, etc.). And that's funny, because this would imply at first glance that the number of integers is equal to the number of even integers. That's the paradoxical property of infinity.
n: Was infinity's paradoxical nature what made the ancient Greeks so uncomfortable with it?
m: Yes, because of its paradoxical consequences. The paradoxical property I just mentioned is the one I start with, because it's the one that came to be the cornerstone for the contemporary treatment of infinity. However, there are other problems that would perhaps be more relevant to the Greek thinking about it.
For instance, are we going to say that a line is made of infinitely many points? This would suggest that the gradual addition of things of one kind (points) gives rise ultimately to something of a different kind altogether (a line). How do you make this leap over dimensionality? How do you get from something that has no dimension at all (a point) to something that has a real dimension (a line)? In the same way, how do you get from lines to surface? At which stage of the addition will you get there? Infinity is troublesome because it seems to imply that once you can add something up infinitely many times you can actually change something from one kind to a totally different kind. That's one kind of problem.
n: What's another?
m: Well, infinity is also problematic because it seems to be more than anything, and for this reason it is something that you will never reach. That's the famous paradox. How will you ever get from Point A to Point B if there are infinitely many things along the way? To get from A to B, you would first have to reach half the way. But to reach half the way, you'd first have to reach half of half the way, and then to get there, you'd have to reach half of the half of the half, and so on. It would appear that you would never be able even to make a start. There are always stages along the way that you haven't reached. If things can be divided into infinitely many stretches along the way, then there is simply no way you will ever cross from one point to another.
Basically, then, the problem seems to be that whatever we're doing getting from one place to another, counting things, doing things, we're doing finite things. We're running a stadium track in two minutes; we're counting, let's say, within the range of a single book; we're making a computation on a piece of paper. We've got something finite to work with. How can anything finite encompass within it something that is infinite in character? So this seems to be another major problem.
And then there is something that is culturally specific. In the Greek context, especially in the Greek mathematical context, there was an enormous interest in trying to find ratios, harmonies, proportions that govern things. It so happens that the ratio of 2 to 3 is correlated in music with the fifth, the ratio of 3 to 4 is correlated with the fourth, the ratio of 1 to 2 is correlated with an octave. Archimedes was interested in the fact that a cylinder is exactly 3 to 2 relative to the sphere that it encompasses. A parabola is precisely 4 to 3 relative to the triangle that it encompasses. There is a sense that permeated Greek thinking in general that in order to understand things you should find the precise integer numbers that govern them.
n: But infinity involves an infinity of numbers that aren't integers.
m: Right. If you're allowing infinity in, then you're not going to have nice integer numbers that govern the relationships. If you allow numbers to be infinitely precise, if you allow them to differ from one another in infinitesimally small amounts, then you get to the point where, for instance, there are two objects—the side of a square and its diagonal, say—and the ratio cannot be given in integer terms. The ratio is problematic; the ratio cannot be defined, because it is infinitesimally given.
The Greeks accepted that you can't actually find two integers whose ratio is the same as the ratio of the side and a diagonal in a square, or the ratio of the perimeter and diameter in a circle. But this is somehow not nice, not what you want to have. For this reason, infinity is somehow not the thing that you want to have mathematically.
n: This discomfort with infinity lasted until the invention of the calculus. How did that change things?
m: What happens with the calculus is that you find ways to calculate infinitely long series. I've mentioned that in order to get from Point A to Point B, you first have to cross half the way, then half of the half, which is a quarter, then half of the half of the half, which is an eighth. This seems paradoxical, because it looks like we can define it this way: Let's have a series, and the series is 1/2 plus 1/4 plus 1/8, etc., going on to infinity. It would appear that the series that has infinitely many terms should be infinitely large.
What you find with the calculus is that there are ways of dealing with infinitely large objects in some ways that are still finite in other ways. There are ways of calculating with infinitely large objects. To start with, this was done on a rather intuitive basis. It was just a series of observations that things could be done and seem to work properly. It was only in the 19th century that precise techniques for dealing with infinitely large magnitudes emerged.
They found that once you allow yourself to do those things in practice, then you can do as a matter of calculation—real calculation with numbers—things that before you'd done purely by operating with geometrical configurations. You can think not in the qualitative terms of geometrical configurations, but in the quantitative terms of dealing with numbers and series of numbers. That's very powerful, because numbers are very powerful. They are precise and manageable in ways that geometrical configurations are not. Essentially, then, you now had a tool that is much more useful for science.
So what appears to be a quaint, paradoxical realm—the realm of infinite magnitude—is actually very practical. It's something that allows you to extend operations with numbers to any domain whatsoever. That's what happened from the 17th century onward.
n: How did Georg Cantor's set theory refine mathematicians' thinking about infinity?
m: Well, the essence of the calculus is that you deal with infinitely large objects. But you never had to define infinity itself, and you never had to worry about the nature of infinity, primarily because you always dealt with the very same kind of infinity—roughly speaking, the infinity of points making up a line, the infinity of all the real numbers between, let's say, 0 and 1. That's the type of thing they were worried about in the calculus from the 17th century to the 19th century.
But they didn't think about what infinity is, because for one thing they didn't think about what a set is. What is a set? And then what would be the difference between a finite and an infinite set? This is something that Cantor did in the late 19th century. Cantor developed the notion of a set, and the notion of an infinite set, a set that has infinitely many objects.
Then many, many curious things turned up. We found that the paradoxical properties of infinite sets can actually be used to define them, and even more striking—and something that is slightly technical, and for this reason perhaps not something that I can show you, but you'll have to take for granted—even more curious is that the fact that you find that once you have an infinity, an infinite set, you can create another infinite set that is bigger than the infinite set you started with. You can have two infinite sets, one of them being bigger than the other. Actually, because the operation is recursive, you can take a set and create a set bigger than itself, so you can have an entire sequence of infinity. There are infinitely many infinities, stretching all the way up.
This, of course, gave rise to a fascinating field of operating with infinite numbers, which, in fact, are richer and more interesting than finite numbers are. That's the field opened up by Cantor.
n: So is infinity an active field of study for mathematicians today?
m: Oh, infinity is what mathematics is about. Infinity always was what mathematics is about. It's just that there are many different ways of dealing with it. In the Greek context, you dealt with situations that give rise to infinity by transforming them into geometrical representations. So, for example, you effectively dealt with how many lines it takes to fill a certain rectangle by looking at various geometrical configurations. You never called infinity by its name (with a few exceptions, including one important one discovered for Archimedes recently).
In the Scientific Revolution, you did those things with the recognition that you're dealing with infinity, and from the 19th century onwards you began dealing with actual infinite sets—a mathematics of infinity. The kinds of infinity that are allowed into the game define the kind of mathematics you're doing even today. That's the fundamental division between different kinds of mathematics. Do you allow in infinities beyond the first, most simple infinity or not? There are different kinds of mathematics dependent on that. Primarily you're dealing with various situations arising from various different kinds of infinity.
Finite things—yes, of course, there are important fields of mathematics that deal with finite situations, and these are fascinating fields in their own right. But the fundmental thrust of mathematics is dealing with infinite situations.
n: In what other fields besides mathematics does the study of infinity come into play?
m: Well, because time and space are things we often think about as continuous, it follows that it is indeed the case that between any two points of time, between today and tomorrow, or between any two points of space, between "here" and "there," there are infinitely many points. And so to do anything with the physical world—that is, physics, chemistry—you have to use the tools of infinity. This is why the calculus is so central for physics. Infinity is the heart of physics.
Of course, in a way you could say that in a quantum world, which is perhaps where we live, reality is not really continuous but discrete. There aren't really infinitely many points between "here" and "there." There are just many, many, many, many jumps, but not infinitely many.
However, they found between the 17th and 19th centuries all sorts of elegant ways to deal with calculations having to do with space and time that assume that those things are continuous. And they are very powerful ways of calculating things. Generally speaking, it's just not worth it trying to calculate things as if they were made of many, many jumps. It is much easier, and in many cases the only way practically to go on, to think of things as if they were continuous.
So even though it's not necessarily the case that the world really is continuous, it certainly is the only way to calculate things about the world, by assuming that it is continuous. Infinity, even if not necessarily the correct description of the world, is certainly the best tool for dealing with the world. As I said, infinity is the basic tool of physics.
n: Does feeling comfortable with the concept of infinity mathematically help you to feel comfortable with it philosophically?
m: That's a very interesting question. There are philosophers who think that because of the rigorous establishment of the calculus in the 19th century, and because of the rigorous treatment of the concept of infinity in set theory from Cantor onward, the philosophical problem of infinity has been solved.
On the other hand, there are interesting complications there. You would notice that the way in the 19th century in which people found a rigorous, precise way of describing infinity was by saying that actually, there aren't such things as infinitesimally small magnitudes; all we have in the actual world are things that are as small we wish. And then, mathematicians have shown from the 19th century onward, with the assumption of things that are as small as you wish, that you can actually develop anything you have in the world with infinity and everything thrown in. So it was thought that the problem of infinity was solved in a certain way and that really that there are no infinitesimals.
But in the 1960s, it was shown for the first time that you can develop a completely equivalent kind of mathematics with infinitesimals thrown in. You can have exactly the same kind of mathematics with exactly the same kind of consequences, so that you don't lose anything. You don't run into any contradictions with the assumption that there are infinitesimals.
In point of fact, we've got two mathematical theories. Both of them derive the same results, but they've got different philosophical assumptions. They assume the world is made of different kinds of constituents, with infinitely small things, without infinitely small things.
Also, as I've mentioned, there are essentially two kinds of mathematics. One of them is known as intuitionism, which is the less standard one. Intuitionists, unlike other mathematicians, don't want to essentially go beyond the very first orders of infinity. They don't allow infinities of higher order. And they show that within intuitionism, you can develop practically all the mathematics that you need. Of those two different mathematical theories, one allows in a whole bunch of infinities, the other doesn't. Yet they derive pretty much the same kind of results.
Once again you see that the philosophical question—What kind of infinities exist?—is not really answered. It's left as a philosophical puzzle. I think it's fair to say that mathematics shows you what's coherent, but what's actually true about the universe is left a philosophical question. In fact, mathematics has shown us during the 20th century that you can have all sorts of mathematical theories out there that are compatible with all sorts of possible universes. I think the philosophical puzzle remains where it was 2,500 years ago and is likely to remain that way.
N: What about you personally? Do you believe, for instance, that space is infinite?
m: That's the question really: Do things exist?! There are many people who say that we should adopt what's known as Occam's razor—namely, don't assume things that are not necessary. Very often this is taken as an argument in philosophy. If you can produce something without a particular assumption, then don't assume the existence of this particular assumption. If you can do without it, better assume it doesn't exist.
I actually don't see the validity of Occam's razor. I think that things can exist even if they don't serve any purpose. My tendency is to be pluralistic. Yeah, I think infinity is a coherent concept. I think I tend to like it. So yes, I tend to believe that infinities exist—all of them, all the way up.
n: Time and space?
m: Yes, time and space, and infinities that go far beyond that, the infinities of pure, abstract mathematics, which are much, much bigger than whatever you need for the development of space and time. I seem to believe those things exist in some mathematical universe. But it's a matter of temperament. There are people who like the universe big and people who like it small. I'm a pluralist; I like to believe things exist.
Becoming numb
This is true no matter how you approach the concept. Many of us might consider numbers the most sure-footed 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 2infinity is.
Try another tack: huge numbers. When we play with mind-boggling figures we non-math 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.
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 10100, or 1 followed by 100 zeroes, and is the largest named number in the West. The Buddhists have an even more robust number, 10140, which they know as asankhyeya. Just for fun, I'll name a larger number yet, 101000. 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.
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 (Birkhäuser, 1982). 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.
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 paradoxes of 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 strides away.
It wasn't for 2,000 years that Zeno's paradox finally got "solved," for all intents and purposes, by the calculus. Its 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.
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 believe 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?
As with numbers, we can also envision space as being infinitely large. After all, if the universe has a boundary, what's on the other side? We 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 human-made 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 light-years 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 its own red dwarf 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.
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.)
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 mind-bending length of time. One of the gamest attempts to define eternity 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 gosh-darn 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, etc., etc.
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."
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."
For mathematicians, infinity means something completely different than for philosophers. It's not something vague and unapproachable, but rather something with a precise definition that lies at the core of modern mathematics. To explain how this eminently practical form of infinity evolved over time, and to translate for the layman how mathematicians think of it, we approached Stanford University classics historian Reviel Netz. A scholar who discovered that, contrary to belief, the ancient Greeks, through the work of Archimedes, had actually toyed with infinitely large sets, Netz knows a thing or two about mathematical infinity..
n: How do mathematicians define infinity?
m: Something which is equal to some of its parts. That's really the technical definition.
n: Is there a difference between the mathematical concept of infinity and infinity in terms, say, of space or time—the philosophical concept?
m: That's the curious thing. Infinity became a really clear and well-defined quantity mathematically in the late 19th century, which it wasn't before and which makes it rather different from what we ordinarily talk about when we talk about infinity, namely, about something very, very big. In mathematics nowadays, when we think about infinity, we think about a set whose properties are different from those of ordinary sets.
n: Can you explain?
m: Well, the defining property of infinity today is that a set's cardinality [the number of elements in a given mathematical set] is equal to the cardinality of some real subset of that set. The thread that links the technical notion of infinity and the more popular/philosophical notion of infinity is the sense that infinity's hugeness being beyond reach endows it with certain paradoxical properties.
A standard example would be what was taken to be something rather paradoxical, the fact that you can take, say, the number 1 and correlate it with the number 2, take the number 2 and correlate it with the number 4, take the number 3 and correlate it with the number 6. In this way you can gradually, step by step, correlate all integer numbers (1, 2, 3, 4, etc.) with all even numbers (2, 4, 6, 8, etc.). And that's funny, because this would imply at first glance that the number of integers is equal to the number of even integers. That's the paradoxical property of infinity.
n: Was infinity's paradoxical nature what made the ancient Greeks so uncomfortable with it?
m: Yes, because of its paradoxical consequences. The paradoxical property I just mentioned is the one I start with, because it's the one that came to be the cornerstone for the contemporary treatment of infinity. However, there are other problems that would perhaps be more relevant to the Greek thinking about it.
For instance, are we going to say that a line is made of infinitely many points? This would suggest that the gradual addition of things of one kind (points) gives rise ultimately to something of a different kind altogether (a line). How do you make this leap over dimensionality? How do you get from something that has no dimension at all (a point) to something that has a real dimension (a line)? In the same way, how do you get from lines to surface? At which stage of the addition will you get there? Infinity is troublesome because it seems to imply that once you can add something up infinitely many times you can actually change something from one kind to a totally different kind. That's one kind of problem.
n: What's another?
m: Well, infinity is also problematic because it seems to be more than anything, and for this reason it is something that you will never reach. That's the famous paradox. How will you ever get from Point A to Point B if there are infinitely many things along the way? To get from A to B, you would first have to reach half the way. But to reach half the way, you'd first have to reach half of half the way, and then to get there, you'd have to reach half of the half of the half, and so on. It would appear that you would never be able even to make a start. There are always stages along the way that you haven't reached. If things can be divided into infinitely many stretches along the way, then there is simply no way you will ever cross from one point to another.
Basically, then, the problem seems to be that whatever we're doing getting from one place to another, counting things, doing things, we're doing finite things. We're running a stadium track in two minutes; we're counting, let's say, within the range of a single book; we're making a computation on a piece of paper. We've got something finite to work with. How can anything finite encompass within it something that is infinite in character? So this seems to be another major problem.
And then there is something that is culturally specific. In the Greek context, especially in the Greek mathematical context, there was an enormous interest in trying to find ratios, harmonies, proportions that govern things. It so happens that the ratio of 2 to 3 is correlated in music with the fifth, the ratio of 3 to 4 is correlated with the fourth, the ratio of 1 to 2 is correlated with an octave. Archimedes was interested in the fact that a cylinder is exactly 3 to 2 relative to the sphere that it encompasses. A parabola is precisely 4 to 3 relative to the triangle that it encompasses. There is a sense that permeated Greek thinking in general that in order to understand things you should find the precise integer numbers that govern them.
n: But infinity involves an infinity of numbers that aren't integers.
m: Right. If you're allowing infinity in, then you're not going to have nice integer numbers that govern the relationships. If you allow numbers to be infinitely precise, if you allow them to differ from one another in infinitesimally small amounts, then you get to the point where, for instance, there are two objects—the side of a square and its diagonal, say—and the ratio cannot be given in integer terms. The ratio is problematic; the ratio cannot be defined, because it is infinitesimally given.
The Greeks accepted that you can't actually find two integers whose ratio is the same as the ratio of the side and a diagonal in a square, or the ratio of the perimeter and diameter in a circle. But this is somehow not nice, not what you want to have. For this reason, infinity is somehow not the thing that you want to have mathematically.
n: This discomfort with infinity lasted until the invention of the calculus. How did that change things?
m: What happens with the calculus is that you find ways to calculate infinitely long series. I've mentioned that in order to get from Point A to Point B, you first have to cross half the way, then half of the half, which is a quarter, then half of the half of the half, which is an eighth. This seems paradoxical, because it looks like we can define it this way: Let's have a series, and the series is 1/2 plus 1/4 plus 1/8, etc., going on to infinity. It would appear that the series that has infinitely many terms should be infinitely large.
What you find with the calculus is that there are ways of dealing with infinitely large objects in some ways that are still finite in other ways. There are ways of calculating with infinitely large objects. To start with, this was done on a rather intuitive basis. It was just a series of observations that things could be done and seem to work properly. It was only in the 19th century that precise techniques for dealing with infinitely large magnitudes emerged.
They found that once you allow yourself to do those things in practice, then you can do as a matter of calculation—real calculation with numbers—things that before you'd done purely by operating with geometrical configurations. You can think not in the qualitative terms of geometrical configurations, but in the quantitative terms of dealing with numbers and series of numbers. That's very powerful, because numbers are very powerful. They are precise and manageable in ways that geometrical configurations are not. Essentially, then, you now had a tool that is much more useful for science.
So what appears to be a quaint, paradoxical realm—the realm of infinite magnitude—is actually very practical. It's something that allows you to extend operations with numbers to any domain whatsoever. That's what happened from the 17th century onward.
n: How did Georg Cantor's set theory refine mathematicians' thinking about infinity?
m: Well, the essence of the calculus is that you deal with infinitely large objects. But you never had to define infinity itself, and you never had to worry about the nature of infinity, primarily because you always dealt with the very same kind of infinity—roughly speaking, the infinity of points making up a line, the infinity of all the real numbers between, let's say, 0 and 1. That's the type of thing they were worried about in the calculus from the 17th century to the 19th century.
But they didn't think about what infinity is, because for one thing they didn't think about what a set is. What is a set? And then what would be the difference between a finite and an infinite set? This is something that Cantor did in the late 19th century. Cantor developed the notion of a set, and the notion of an infinite set, a set that has infinitely many objects.
Then many, many curious things turned up. We found that the paradoxical properties of infinite sets can actually be used to define them, and even more striking—and something that is slightly technical, and for this reason perhaps not something that I can show you, but you'll have to take for granted—even more curious is that the fact that you find that once you have an infinity, an infinite set, you can create another infinite set that is bigger than the infinite set you started with. You can have two infinite sets, one of them being bigger than the other. Actually, because the operation is recursive, you can take a set and create a set bigger than itself, so you can have an entire sequence of infinity. There are infinitely many infinities, stretching all the way up.
This, of course, gave rise to a fascinating field of operating with infinite numbers, which, in fact, are richer and more interesting than finite numbers are. That's the field opened up by Cantor.
n: So is infinity an active field of study for mathematicians today?
m: Oh, infinity is what mathematics is about. Infinity always was what mathematics is about. It's just that there are many different ways of dealing with it. In the Greek context, you dealt with situations that give rise to infinity by transforming them into geometrical representations. So, for example, you effectively dealt with how many lines it takes to fill a certain rectangle by looking at various geometrical configurations. You never called infinity by its name (with a few exceptions, including one important one discovered for Archimedes recently).
In the Scientific Revolution, you did those things with the recognition that you're dealing with infinity, and from the 19th century onwards you began dealing with actual infinite sets—a mathematics of infinity. The kinds of infinity that are allowed into the game define the kind of mathematics you're doing even today. That's the fundamental division between different kinds of mathematics. Do you allow in infinities beyond the first, most simple infinity or not? There are different kinds of mathematics dependent on that. Primarily you're dealing with various situations arising from various different kinds of infinity.
Finite things—yes, of course, there are important fields of mathematics that deal with finite situations, and these are fascinating fields in their own right. But the fundmental thrust of mathematics is dealing with infinite situations.
n: In what other fields besides mathematics does the study of infinity come into play?
m: Well, because time and space are things we often think about as continuous, it follows that it is indeed the case that between any two points of time, between today and tomorrow, or between any two points of space, between "here" and "there," there are infinitely many points. And so to do anything with the physical world—that is, physics, chemistry—you have to use the tools of infinity. This is why the calculus is so central for physics. Infinity is the heart of physics.
Of course, in a way you could say that in a quantum world, which is perhaps where we live, reality is not really continuous but discrete. There aren't really infinitely many points between "here" and "there." There are just many, many, many, many jumps, but not infinitely many.
However, they found between the 17th and 19th centuries all sorts of elegant ways to deal with calculations having to do with space and time that assume that those things are continuous. And they are very powerful ways of calculating things. Generally speaking, it's just not worth it trying to calculate things as if they were made of many, many jumps. It is much easier, and in many cases the only way practically to go on, to think of things as if they were continuous.
So even though it's not necessarily the case that the world really is continuous, it certainly is the only way to calculate things about the world, by assuming that it is continuous. Infinity, even if not necessarily the correct description of the world, is certainly the best tool for dealing with the world. As I said, infinity is the basic tool of physics.
n: Does feeling comfortable with the concept of infinity mathematically help you to feel comfortable with it philosophically?
m: That's a very interesting question. There are philosophers who think that because of the rigorous establishment of the calculus in the 19th century, and because of the rigorous treatment of the concept of infinity in set theory from Cantor onward, the philosophical problem of infinity has been solved.
On the other hand, there are interesting complications there. You would notice that the way in the 19th century in which people found a rigorous, precise way of describing infinity was by saying that actually, there aren't such things as infinitesimally small magnitudes; all we have in the actual world are things that are as small we wish. And then, mathematicians have shown from the 19th century onward, with the assumption of things that are as small as you wish, that you can actually develop anything you have in the world with infinity and everything thrown in. So it was thought that the problem of infinity was solved in a certain way and that really that there are no infinitesimals.
But in the 1960s, it was shown for the first time that you can develop a completely equivalent kind of mathematics with infinitesimals thrown in. You can have exactly the same kind of mathematics with exactly the same kind of consequences, so that you don't lose anything. You don't run into any contradictions with the assumption that there are infinitesimals.
In point of fact, we've got two mathematical theories. Both of them derive the same results, but they've got different philosophical assumptions. They assume the world is made of different kinds of constituents, with infinitely small things, without infinitely small things.
Also, as I've mentioned, there are essentially two kinds of mathematics. One of them is known as intuitionism, which is the less standard one. Intuitionists, unlike other mathematicians, don't want to essentially go beyond the very first orders of infinity. They don't allow infinities of higher order. And they show that within intuitionism, you can develop practically all the mathematics that you need. Of those two different mathematical theories, one allows in a whole bunch of infinities, the other doesn't. Yet they derive pretty much the same kind of results.
Once again you see that the philosophical question—What kind of infinities exist?—is not really answered. It's left as a philosophical puzzle. I think it's fair to say that mathematics shows you what's coherent, but what's actually true about the universe is left a philosophical question. In fact, mathematics has shown us during the 20th century that you can have all sorts of mathematical theories out there that are compatible with all sorts of possible universes. I think the philosophical puzzle remains where it was 2,500 years ago and is likely to remain that way.
N: What about you personally? Do you believe, for instance, that space is infinite?
m: That's the question really: Do things exist?! There are many people who say that we should adopt what's known as Occam's razor—namely, don't assume things that are not necessary. Very often this is taken as an argument in philosophy. If you can produce something without a particular assumption, then don't assume the existence of this particular assumption. If you can do without it, better assume it doesn't exist.
I actually don't see the validity of Occam's razor. I think that things can exist even if they don't serve any purpose. My tendency is to be pluralistic. Yeah, I think infinity is a coherent concept. I think I tend to like it. So yes, I tend to believe that infinities exist—all of them, all the way up.
n: Time and space?
m: Yes, time and space, and infinities that go far beyond that, the infinities of pure, abstract mathematics, which are much, much bigger than whatever you need for the development of space and time. I seem to believe those things exist in some mathematical universe. But it's a matter of temperament. There are people who like the universe big and people who like it small. I'm a pluralist; I like to believe things exist.
