Infinity

Ken Taylor
Welcome to Philosophy Talk, the program that questions everything…

John Perry
…except your intelligence. I’m John Perry.

Ken Taylor
And I’m Ken Taylor. We’re coming to you from the studios of KALW San Francisco.

John Perry
Continuing conversations that begin at Philosopher’s Corner on the Stanford campus.

Ken Taylor
Today: Infinity. Philosophers, theologians, and students of nature have had intimations of the infinite everywhere in the all-knowing mind of God.

John Perry
In the beginningless past and the endless future.

Ken Taylor
In space without boundary.

John Perry
In matter that can be divided into smaller and smaller bits without limit.

Ken Taylor
And in the realm of pure number.

John Perry
We’ll begin our exploration of the infinite by sampling what has been thought and said about infinity in the past.

Ken Taylor
Then we’ll dig into some puzzles and paradoxes raised by the very idea of the infinite. Are there actual complete infinities? Are all infinities only potential? Can infinities come in different sizes? Does space have an outermost boundary? Does time have a beginning or an end?

John Perry
We’ll close with a discussion of the experience of the infinite. We’ll examine the representation of the infinite in art, literature, mathematics, and science, and we’ll ask how our finite, feeble minds can ever grasp the concept.

Ken Taylor
So, John, let’s start with the definition. How would you define infinity?

John Perry
Piece of cake, Ken. The infinite is that which is not finite. Show’s over. Let’s go home. Thanks for thinking, folks.

Ken Taylor
Wait a minute. Wait a minute. You’ve only told me what infinity is not. That doesn’t tell me anything positive and definite about infinity. Suppose I ask you to define the color green by analogy, and you said green is the color that is not red, not blue, not orange, and so on for every other color you can think of. That would tell me a lot about what green is not, but it doesn’t tell me much at all about what green is. I don’t want to know what infinity is not. I want to know what it is. I want a concrete, positive definition to help me recognize infinity when I come across it.

John Perry
I hate to tell you this, Ken, but you’re not very likely to come across it. Every number you’ve ever counted to has been finite. Every extent of space you’ve ever crossed has been finite, and every span of time you’ve ever experienced has been finite too. And every thought you think is finite, because you can have only a finite mind.

Ken Taylor
Okay, I grant you have a finite mind, but what’s your point? I don’t get it.

John Perry
Well, take your color analogy and apply it to the case of number. Then we could define infinity like this: infinity is the number that is larger than any finite number you can think of. Infinity is not one, not two, not 87, not a million. It’s take the number of the national debt. Even that is not infinite. What is it? What is infinity? It’s precisely the number that is not any of those finite numbers and is larger than every single one of them.

Ken Taylor
But that definition’s worse than my negative definition of green. I mean, there are only so many colors, so you could pick out all but one and say green is the missing one. And when you came across a color that wasn’t any of the ones you already knew, you could say, “Aha! There it is. That must be green, but you can’t do that with infinity and the numbers because you’re never going to run out of numbers and you’re never going to get to the last but one number.

John Perry
In spite of yourself, Ken, you’re starting to get it. You’re starting to understand the concept of infinity.

Ken Taylor
I’m lost. What do you mean? I don’t I don’t get it.

John Perry
Well, ask yourself how many numbers there are.

Ken Taylor
Lots. There are lots and lots of numbers.

John Perry
There are more than lots of numbers. There are an infinite number of numbers, and what this shows, or rather, what shows this, is the very fact that you cannot exhaust them by running through them one by one. Just try. I’ll give you as much time as you want.

Ken Taylor
Wait a minute. Are you? I think I get it. You’re equating infinity with inexhaustibility. Is that your positive and definite definition of the infinite? Something you can’t run through, something you can’t exhaust?

John Perry
Well, it’s a good start, but it’s not all there is. Infinity has lots of other cool properties. Like,

Ken Taylor
give me, give me some more.

John Perry
Well, for example, if you take half of infinity, you get infinity again. I mean, just take the the even numbers, right? So, so that’s. Seems like it’s half of the numbers, but you’ll never run out of even numbers too, so they’re infinite. So if you double infinity, guess what?

Ken Taylor
You get infinity again.

John Perry
Absolutely. So there’s something else positive indefinite to say about infinity. Infinity is that amount which you can’t decrease by dividing it, and you can’t increase by multiplying it.

Ken Taylor
Okay, so that makes infinity that which there is nothing greater than, right?

John Perry
No, not quite. Infinities come in different sizes. Some larger, some smaller. As far as we know, there’s an infinity of infinities.

Ken Taylor
Wait, wait a minute. Wait a minute. Something can’t be right here. You just said you can’t increase infinity by multiplying it. You can’t decrease it by dividing it. So how could there be different sizes of infinity?

John Perry
Well, I’m going to have to leave you hanging there, Ken. It’s time to hear from some other voices. Our roving philosophical reporter Zoe Corneli spoke to a number of students about their understanding of infinity. She files this report

Zoe Corneli
After a long day of classes at San Francisco State University, students relax in lounges and cafes, it’s the perfect time for a philosophical conversation. Theater major Tatiana Rivas takes a break from chatting with friends to share her thoughts about infinity.

Tatiana Rivas
If I think about it, the only thing that will come to mind is life. I don’t think life stops. Maybe your body, yet physically, you you know you’re not living, breathing, or whatnot. But I feel like people still remember who you are, and I think it kind of relates to different things like space. Physically, this building looks like it just stops right here, like that corner, that corner, the corner over there. It’s just something in the way, but push it down. What are you gonna have space?

Zoe Corneli
Jamie Tretheway, a physical education student, has a slightly more cynical point of view.

Jamie Threthaway
When I think of infinity, I think of school because it’s never ending, and because I’m still in school and I’m 24, and I remember when I was a sophomore in high school, and I remembered thinking, three more years and I’m done with school forever. And now, 10 years later, I’m still in school.

Zoe Corneli
Tim Schultz is a high school senior visiting the campus for a meeting. To him, infinity takes on an altruistic meaning.

Tim Schultz
I feel that infinity has a has a relationship to helping others, and I feel that when you help others, there is no limit, and it should always be ongoing.

Zoe Corneli
Social work graduate student Sarah Bennett remembers the first time she heard the term. When

Sarah Bennett
I think of infinity, I think of kids who say use the word infinity, like I love you, infinity, or I’ll be your best friend, infinity, and that’s the first like reference that I ever had to that word. Being a kid trying to figure out what it meant, like what’s the most the longest thing you could ever think of something that’s so big that there’s no number for it. That’s the way it was described to me as a kid, and it was so hard to imagine.

Zoe Corneli
Nearby, education students Keith Favero and Antoine de Friedman are engaged in a heated political discussion. I interrupt to ask them what they think about infinity. Favero is surprised to learn his friend doesn’t spend much time worrying about it.

Keith Favero
You’ve never thought about how frightening it is! Do I want to use the word frightening? It is that that something is limitless, because I’ve laid in bed at night thinking, you know, this world is flying through space at X number, however many 1000s of miles per hour. I’m thinking, you know, I’m sitting here in this, lying here in this bed. My God, where am I going? Yeah, I’ve thought about that. It’s creepy to me.

Antoine de Friedman
The thing is that whatever happened up there doesn’t really change my life here now. What I do and the fact that I have to go and wake up tomorrow and go to work.

Zoe Corneli
Favero insists there’s more to life than the daily grind.

Keith Favero
You have to go get something to eat and go to bed and get up and go to work. And okay, those are the day-to-day things. But still, that we’ve got this other part of us, these nagging questions that never go away, and they’re infinitely with us, you know.

Zoe Corneli
De Friedman admits infinity can show up in our day-to-day lives.

Antoine de Friedman
I mean, our creativity is limitless. I mean..

Keith Favero
I don’t agree with that.

Antoine de Friedman
Why not?

Keith Favero
Because a human creativity, we we have a finite amount of, you know, cognitive.

Zoe Corneli
Now this conversation could go on forever, but we’ll leave it there for now. For philosophy talk, I’m Zoe Corneli.

John Perry
I’m John Perry, and with me is Ken Taylor.

Ken Taylor
Our guest is Rudy Rucker. He’s professor emeritus of computer science from San Jose State University. He’s not just a professor of computer science; he’s a mathematician, a philosopher, a novelist. He’s author of more than 30 books, I think, and most not most recently, but relevant to our subject today, Infinity and the Mind. Rudy, welcome to Philosophy Talk.

Rudy Rucker
It’s nice to be here,

John Perry
Rudy. You’ve you’ve written both. Fiction and nonfiction books about the infinite. When and why did you first become fascinated or even obsessed with infinity?

Rudy Rucker
Actually, one of the guys you had on, I had that experience of being amazed that the universe goes on forever or it might go on forever, and just wondering what lay beyond. And in high school, I think I read a book of mathematics popularization, and it mentioned that there’s different levels of infinity, and that that intrigued me very much. That we have the natural numbers has one size, and then the the number of points in space is supposedly larger, and that that struck me as a very strange fact. And then I went to graduate school, and the it was the 1960 s then, and the notion of infinity seemed very very natural. I was interested in enlightenment and seeing God, and so I ended up majoring in set theory, which is the mathematical study of infinity.

Ken Taylor
You majored in set theory because you thought about enlightenment and seeing God. That’s that’s kind of yeah. It’s kind of surprising.

Rudy Rucker
I think you could really say that set theory is mathematical theology.

John Perry
So so theology goes way back. Does infinity go way back? Who’s who are the first ones to think about it, and what did they say?

Rudy Rucker
Well, that’s the history of infinity is quite interesting. The Greeks viewed it in a sort of negative sense. We very often go back to the Greeks when we talk about history of mathematics, and Pythagoras had thought everything in the world could be expressed in terms of relations between natural numbers, like a ratio, and they were disturbed to discover that if you take the length of the diagonal of a square, you draw a square and you take the length of its diagonal, you can’t express that as a ratio of two natural numbers. And then there is a way to describe that length, but the description is infinite, and that that sort of freaked them out.

John Perry
So that’s what we call a real number, like pi. If I mean, you could just go on forever.

Rudy Rucker
Or the square root of two, and we’re used to right. We have this convention now. We say, oh, that’s a decimal. We put point, and then 31415977, and we’re sort of used to the notion that that goes on forever. But that whole notion of defining numbers that way didn’t really come into fashion until about 100 years. So you

Ken Taylor
said, but you said the the Greeks didn’t think of. You said something about how the Greeks thought it. They thought of infinity negatively. Yeah, they found this result right, which like that’s an that’s a real that’s an actual number of some sort. But why why is it negative? Because if you find this result that you can’t express this by a ratio, a finite ratio, right? But you say there is really a number. How is that thinking of it negatively? I don’t quite.

Rudy Rucker
Well, they thought of it as a privation, as a the the word for infinite was aperon, meaning not having a limit or being formless. They they thought of it as being something like a crumpled handkerchief instead of some nice smooth like curve like a sphere. So it was just messy and scuzzy, and they didn’t view it as as being good. So

Ken Taylor
that was an ugly number, an ugly result. Yeah. But now

John Perry
somewhere along the line in the Middle Ages, being infinite became a good thing because God’s supposed to be infinite. Were the were these guys thinking about the problem that Pythagoras had when they said God’s infinite?

Rudy Rucker
Well, there’s there’s some interesting discussions among the theologians in the Middle Ages. They they became curious, could God make something that was infinite, and they thought maybe you couldn’t have something that was infinite. That thing you mentioned earlier that there’s as many even numbers as there are regular numbers. In the the medieval mind, they thought maybe that meant that there was something contradictory about being infinite, and so they said God couldn’t make anything infinite. And then Augustine said, “Who are? What are you mean wretches to presume to limit His power? Right,

Ken Taylor
right. Good

John Perry
old Augustine.

Ken Taylor
Augustine. I mean, Augustine was into the infinity of God and the. I think the infinity of time, but the kind of eternal. God’s supposed to be outside of time, right? Because there’s two different kinds of it. Yeah, time is everlasting, but God’s outside of it, or something like that.

Rudy Rucker
Well, there’s one notion that comes up. It’s in set theory they call it the reflection principle, and that God should be unknowable. So if if God was the only infinite thing, we could say, okay, we can define God. God is the the thing that’s infinite, and so there’s a feeling that there ought to be some things less than God that are also infinite. Otherwise, we would be able to put God in too small of a box.

Ken Taylor
Well, let’s explore that a little more fully. We’re just getting started with this. You’re listening to Philosophy Talk today. We’re discussing infinity with Rudy Rucker, author of Infinity of the Mind and 29 Other Books.

John Perry
We started by examining some ancient conceptions of the infinite. But coming up, we’re going to dig into modern conceptions and the puzzles and paradox that seem to come out of the infinite and, to some extent, motivate the study of the infinite. Are there actual complete infinities, or are all infinities only potential? Can infinites really? Come in different sizes. Does space have an outermost boundary? Does time have a beginning and end? Have you ever been puzzled by the concept of infinity yourself? Join us by calling toll free at 1-800-525-9917. That’s 1-800-525-9917.

Ken Taylor
Pondering the infinite plus your calls and emails when philosophy talk continues.

John Perry
How do you know when you’ve reached infinity? How is the infinite distinguished from the finite? Are there real concrete infinities in nature, or is the infinite merely a projection of our minds? This is philosophy talk, and I’m John Perry,

Ken Taylor
and I’m Ken Taylor. Join our discussion and help us ponder the infinite. The toll-free number 1-800-525-9917. I guarantee you that won’t take you an infinite amount of time to dial or email us at comments@philosophytalk.org.

John Perry
Our guest is Rudy Rucker, author of Infinity and the Mind.

Ken Taylor
So, Rudy, you were talking about how the Greeks were puzzled about the concept of infinity, and this result they thought was ugly and degenerate. And part of what went with that is people, for a long time, I think until really the 19th century, when they thought about infinity, they thought it had to be merely potential, right? Because you could keep going on forever. You could never have an actual complete infinity. Right. Help us understand this distinction between actual and potential infinity.

Rudy Rucker
That was, I think, Aristotle who talked about that, and the idea being, if I start counting 123456, seven.dot.so, then you say, okay, I have a potentially infinite set, and then some people would say, but you could never round it off. And in around 1880, Georg Cantor started saying, we can treat it as a finished object. And then there was debate. Well, you don’t see it in the physical world, and though maybe we do have some physical infinities we’re not aware of, Cantor would say, and he would also say even if there’s not a physical infinity, we have this sort of ideal world of metal objects, and in there,

Ken Taylor
let’s stop for a second. Just just stop for a second. Go a little bit more slowly, because you know, you went from Aristotle to Cantor, but there’s a lot in between. Like Kant also thought that the idea of an infinite totality, of an actual collected totality-that’s an infinite totality, that’s complete and finished-was deeply incoherent. I mean, did Cantor have some insight that people like Kant and Aristotle were missing?

Rudy Rucker
I think the difference was, it seemed the the properties of an actual infinite set are somewhat paradoxical, and for for a time it was believed that it was incoherent or contradictory, and one of the classic examples of that is the fact is that you can have an infinite set be in a one to one correspondence with a subset of itself,

John Perry
so so that’s the idea we explored a little. That, that I mean, I was spoon-fed a little set theory on my way to getting a Ph.D. in philosophy. So, as I get it, the idea is that infinity is, as studied now, is mostly a concept, a property of sets, of collections. So you take the the set of even numbers, and you take the set of numbers. I mean, natural numbers, 123, and somehow by some kind of flim flam, the set theorist says there’s as many even numbers as there are numbers. How can that possibly be right?

Rudy Rucker
Well, flim flam is what set theory is all about. There’s a there’s an even simpler example of the the paradox you’re talking about, and David Hilbert called it Hilbert’s Hotel. Suppose we have a hotel with infinitely many rooms, one for each natural number. So you’ve got room one, room two, and so on. And then the hotel is full, and you have a new guest show up, and then the the clerk says, “Well, there’s no room. And the guest, being a set theorist, says, “You can make room for me. Take the guy in room one and put him into room two. Take the guy in room two and put him into room three. Take the guy into room three. And so, after an immense amount of bustle, they they free up room number one. Yeah, right. Because

Ken Taylor
they’re always going to have a room to put the next guy. Since they never going to infinite number of rooms, they just keep pushing everybody over one. Yeah,

John Perry
so that

Ken Taylor
so if if 10 guests, if 100 guests, if an infinite number of guests showed up, you could still yeah find a place. See,

John Perry
but then now my my reaction to that is well, that shows there’s no infinite hotel, and probably serious people shouldn’t waste their time worrying about no

Ken Taylor
to put it back in these old terms. There’s no actual complete infinity. That is the infinite hotel. There’s just

Rudy Rucker
well, that’s the sort of pre-Kantorian way of looking at things. But I guess the burden was upon mathematicians to show that.

John Perry
Two sets that met that definition, but they couldn’t be put into one one-to-one correspondence, and you’d have shown that there’s two infinities. Very clear. But okay, that’s as far as I get. I can’t think of any. What is a good example?

Rudy Rucker
Well, if think of the the real numbers, think of the number, the points on the number line between zero and one. So each of those I could write as a decimal of this form 0.17 whatever you know or 0.3333, and so then say could I list those things in a list matching the natural numbers a first one a second one a third one, and Cantor formulated the so-called diagonal argument, and he said suppose you make take a list of these decimal numbers, you can always find a number that’s not in the list, and the trick is involves going down the diagonal. Make sure it differs from the first one in the first decimal place, the second one in the second decimal place, the third in the third decimal place, and you can whip out there and you come up with a number that’s not in the list, and you can always do this. Therefore, the set of real numbers is larger than the set of natural numbers.

John Perry
So, so this little bit of hocus pocus, legerdemain, yeah, legerdemain goes back to Pythagoras. So, so history is kind of coming around. I mean, because it’s that the fact that you take an endless number of digits to write out pi is what makes this argument work.

Rudy Rucker
Exactly, and that you can sort of see that’s why it’s infinity’s building on itself.

Ken Taylor
So 1-800-525-9917 or email us at comment@philosophytalk.org. Now I want I want to pressure at a different kind of point. Change the subject slightly. I see. I understand the mathematical concept of infinity. I spent a fair amount of time digging into that in my youth and being fascinated by the set theory and large cardinals and all that, right? I understand the kind of religious motivation, right? You think the world is a finite thing; it’s got to have a source. It’s got to have a source that’s not finite because you know, and you think about time. But here’s a question for you: Is infinity just a mathematical or religious concept? Are there really any real physical infinities? I mean, you might think there must be one-the infinite divisibility of space or something. So there’s an infinitely small point in space, or the infinite extendability of space, or something. But that’s an area

Rudy Rucker
that’s always fascinated me. It’s like we’ve got this beautiful science of the infinite, and the physicists haven’t latched onto it. And you mentioned the trouble is quantum mechanics. I remember the the mathematics popularizer Martin Gardner. I was once talking to him, and he said sadly, quantum mechanics ruins everything. And when you want to talk about infinite divisibility of matter. Then they say, “Well, once you get down to 10 to the minus 43rd meters, space turns into this this foam, and it doesn’t even make sense to talk about dividing it. And that’s sort of a killjoy attitude. And time

Ken Taylor
doesn’t go infinitely back. Well, we’ve got a bridge as a big bang.

Rudy Rucker
Well, the cosmologists-it’s amazing how rapidly they change their minds. I mean, you think everything’s nailed down, and then you you stop paying attention for five years, and you come back, and and now they say the the the universe is expanding forever. So time might go on forever. And it used to be that they said if the universe expands forever, that means there’s infinitely many stars. So that’s they’re not quite saying that yet, but that seems to be back on the table. So I

Ken Taylor
don’t know. Are there infinities in the physical world or not? Are they only in the mathematical world or in the spiritual world and not in the physical world?

Rudy Rucker
Well, I’m I’m optimistic about it. I think the the the Planck limit down at 10 to minus 33rd meters is maybe just a sort of something that we can go through. I’d like to imagine that there’s infinitely many levels of matter beneath there, so we could would have infinite divisibility.

Ken Taylor
We’ve got callers on the line 1-800-525-9917, or email us comments@philosophytalk.org. And Bill in San Francisco, welcome to Philosophy Talk, Bill.

Bill
Hi, I just have a simple question: If you could compare, contrast, or what is the difference between zero and infinity? If you multiply two times zero, you still get zero. Multiply two times infinity, you still get infinity.

Rudy Rucker
That’s a cute question. Intellectually, sometimes I also think of thinking about infinity as a form of meditation, and there’s two schools of meditation where one is where you try to visualize zero, and the other is where you try to visualize infinity. That is, you try to throw everything out of your mind and think nothing at all, or you try to encompass the entire cosmos. And there’s a certain duality there, and one over infinity is zero, so they’re they’re friends from way. Yeah, yeah,

Ken Taylor
but zero is not larger than every finite number, so there’s a big difference. Yeah, well, not

Speaker 7
that every finite number. Okay, John, you

Ken Taylor
got an email there, right? Yeah,

John Perry
Claudia wants to talk a little more about the hotel example. She says it occurs to me that in your hotel example. The last customer, the room in number room number infinity would not have to change rooms. Isn’t room infinity plus one the same as room infinity, or maybe they’re adjoining rooms?

Rudy Rucker
Well, if we can have a in my novel White Light, I wrote about a mountain that goes on past every level of infinity, and we could actually, you could squeeze before room infinity. All the hustle and bustle would be down there, and you’d fit the other person in. But then the guy at room infinity wouldn’t have to be disturbed. But then, of course, there’s an infinite sequence of rooms beyond that. There’s room infinity plus one, infinity plus two. There’s there’s different. It’s a subtle distinction.

John Perry
It’d be a long time before he would be disturbing.

Rudy Rucker
Well, if you want to do things, you want to take advantage of Zeno, and you do the first move in half a second, the second move in a quarter of a second, the third move in an eighth of a second. Did

Ken Taylor
we start out with nobody in in room infinity plus one? Is that how this went when we filled up all the rooms? Well, we don’t have to

Rudy Rucker
bother that guy. Yeah. So

John Perry
you mentioned Zeno. Maybe maybe we should think of at least one of Zeno’s paradoxes to make the show kind of melt round. Sure, yeah. I should I should have

Rudy Rucker
talked about Zeno. And and his famous example is that you can fit an infinite sequence into a finite interval. Like to get out to get to the door, you go halfway there, then you go half the remaining distance. Then you go half the remaining distance, and you actually have to carry out an infinite sequence of actions to get there. But yet you’re able to do it, and the trick is that each of those successive actions is done twice as fast as the one before.

Ken Taylor
But his mistake was, I think, that he thought that if you summed an infinite number of things, you had to get infinity, right? So there are infinite number of smaller and smaller intervals, and he thought the sum of those had to be infinity. And he didn’t understand the notion of a limit. But maybe we can talk about that more after the break. You’re listening to Philosophy Talk. We’re discussing infinity with Rudy Rucker from San Jose State University.

John Perry
We’ve been exploring the nature of infinity. Next, we’re going to dig into the experience, cognition, and representation of infinity in areas as diverse as art, literature, science, religion, and mathematics. Can our feeble finite minds really grasp, experience, or even explain the infinite? Experiencing and explaining the infinite when philosophy talk continues. What would it be like to experience the infinite? How can our feeble finite minds even hope to understand, let alone explain the infinite. I’m John Perry. This is Philosophy Talk, the program that questions everything

Ken Taylor
except your intelligence. Our guest is Rudy Rucker, author of Infinity of the Mind and lots of other books.

John Perry
Rudy, we’ve been talking a lot about the nature of the infinite, but now let’s ask about our experience and cognition of the infinite. How can small, limited brains like you know? We say we question everything except our listeners’ intelligence, but we keep saying they got finite minds. But anyway, putting that aside, how can our small, limited brains possibly grasp the nature of this elusive thing we call the infinite?

Rudy Rucker
Well, one example you might think of is in perspective. When you do a perspective painting, you can actually represent an infinite railroad track by having it dwindle to this point on the horizon. So we we do things like that. We have we sort of summarize the infinite concept by a finite point.

Ken Taylor
In my science fiction, sometimes I try to write about people that are in that are doing infinite things, and I tend there very often to fall back on what I call the Zeno speed up, where they can do, as we were discussing before the break, they can do one thing in half a second, the next thing in a quarter of a second, the next in an eighth, and as Ken mentioned, that actually sums only to a second. It’s not really a paradox. It’s just strange. Yeah, yeah, but but it sounds-is it? Do you think the case that in the 19th century of Cantor and since then we’ve unders we’ve grasped the infinite in a new way? Because you think about all the other-it’s back to this potential versus indefinite. Even your perspective painting-I mean, that’s a represent-that’s a representation of something not terminating and going to a terminus, but never reaching there. I mean, is that the only way we can really grasp the infinite? You know, I mean, John keeps calling Cantor magical or something or sleight of hand when you say, ah, take the totality. But does that really make any sense? Well,

Rudy Rucker
I think it would be nice if the physicists came through for us and we started saying, well, there’s different levels of the transfinite. There’s Cantor made up these wonderful names for them. He used the Hebrew alphabet because I think it sounds sort of cabalistic and mysterious. And the first letter of the alphabet is Aleph, and so he has Aleph null, and then Aleph one, and Aleph two, and it would be very cool. If you could say something like an electron is an aleph one condensation point, yeah, proton is an alpha two point. You can’t.

Ken Taylor
So if I, you know, we’ve got a caller on the line who wants to ask us about a kind of number. I’ve heard of lots of different numbers, transfinite cardinals in imaginary numbers, real numbers. Philip in San Francisco, welcome to Philosophy Talk.

Philip
Hi, hi. I want to talk to Rucker. I read this article years ago in Magazine. I’ve forgotten the core of it, but I remember this was really gave me a strange feeling that stayed with me, and it was about surreal numbers, and it was about infinite infinity and the surreal with those particular numbers, and it also said there’s plenty of things we need numbers like these for, and other kinds of numbers like them. And I like to ask him that. Basically, what are surreal numbers, and why do we need more and other kinds of numbers? Haven’t we got plenty already? Thanks for the call,

Rudy Rucker
Philip. Yeah, those were they were invented by the mathematician John Horton Conway, who’s also famous for inventing the the game of life, computer graphic, and Donald Knuth, a computer scientist, helped him popularize this notion. And in a nutshell, the idea is we have Cantor’s transfinites like Aleph zero, Aleph one, Aleph two, Aleph seven, and then Conway and Knuth said, “Well, let’s let’s beef up our number system so we can take the reciprocals of them. So rather than instead of just saying one over infinity is zero, let’s say one over aleph null is one tiny little number. One over aleph one is a slightly smaller number. One over aleph two is even smaller.

Ken Taylor
Are they smaller than like the smallest real? I mean, yeah, the the

Rudy Rucker
smallest real number you you have like point oh point oh oh point oh oh one, you’d sort of have maybe aleph null zeros and then a one or a ones.

Ken Taylor
My mind’s getting boggled here, John. You got an email. Yeah, we got

John Perry
another interesting email. This is Mike from Pasadena, and he says infinity is by definition something we cannot fully perceive, and yet we have a word for it. But I wonder if having a word for something really implies our knowledge of it, or is it ultimately a delusion?

Ken Taylor
There are people who who are finite math and finitists who think that these notions of infinity should be banished from mathematics. Well, George Cantor

Rudy Rucker
was always fighting people like that. He says it’s a it’s a form of myopia or short sightedness that destroys the possibility of seeing the truth. You’re afraid of something, so you just you just shout that it doesn’t exist, and you refuse to look out the window. But if you let set theory open your mind, you can see these things very clearly.

John Perry
So now, now you you obviously love the infinite. Oh yeah, but but you are at least by profession your day job for a long time was being a computer scientist. That’s right. You try to come down from from infinity to

Rudy Rucker
256k. Now the the

John Perry
physicists haven’t haven’t done what you want and really recognize the necessity for a real infinite thing. How about computer science? Does it need the infinite, and how’s it going to get it?

Rudy Rucker
Well, it’s sometimes we dream about new kinds of computer science. This like quantum computation, and then or people talk about what if we can build a computer that can travel back in time. There’s an interesting article in the Scientific American this month about fantastic types of computers, and certainly if we could have a computer that could use Zeno speedups and do an infinite number of steps.

John Perry
So, in other words, you get Intel to build a computer or chip that does the first instruction and then the second instruction half that time. Exactly.

Rudy Rucker
That sounds. If you got a patent on that, well, see, then instead of having to write my next novel, I simply feed in the previous 16 and have it search through the all possible novels to find one that would appear to be a reasonable successor.

Ken Taylor
Some religious people think that they have a kind of experience of the infinite in contemplating the universe in awe and wonder. I’m not quite sure what it would be like to experience the infinite, but lots of religious traditions have this idea.

Rudy Rucker
Yes.

Ken Taylor
What What do you make of that idea?

Rudy Rucker
Well, in the 1960s, we called it the white light. It was this this thing of fusing into this this blinding white light of mystical enlightenment, and whether or not it was potentiated by by meditation or more worldly means, the feeling was there’s something there. And at various times in your life, you will have this sensation of a union with with the cosmos at the very least, and that’s sort of one of the driving kind of emotional reasons that that keeps me at least interested. Infinity is this idea that I’ve always been afraid of dying from from my earliest years, and the hope that there’s something beyond that that I can connect with, and

Ken Taylor
so that’s that’s sort of maybe your mathematical explorations of infinity. I’m just had this thought for the first time. When you really dig into these complicated technical notions of infinity, maybe they help prepare the way somehow to experience the infinite. I mean, because you might. Think that well, the religious experience of the infinity is one thing, and this mathematical cognition of infinite sets, right? But maybe the one gives real content to the other. What do you What do you think about? Yeah,

Rudy Rucker
I really do have that feeling. At the beginning of the hour, I said in a way, set theory is a an exact form of theology, and it’s it is a way of removing like these notions that it’s I can’t even think about it. It doesn’t make sense. It’s incoherent. And then you do the mathematics. You say, well, there actually could be something there, and it does, as you say, open your mind to the possibility of perceiving the absolute.

John Perry
So on the friends of infinity side, we’ve got an email from Fred Eday, and then I want to add my two cents at the end. Fred says, “Here’s my take on infinity. If a geometrical point is infinitely small, and if all physical points contain geometrical points, therefore all physical objects are infinite, we live in the infinite. It’s all around us for any eye to see. So my follow-up is this: Aren’t these physicists being a little duplicitous? I mean, because they use the calculus, they love the calculus, and and calculus has limits, and isn’t that really borrowing the infinite there?

Rudy Rucker
Oh yeah, the physicists-they’re very-they’re complete sleazebags. Cantor himself said infinity surrounds us, and he actually worked out a theory, which unfortunately doesn’t seem to have been true. But he said, “Well, physical matter is made of aleph zero points, and things like electrical fields are made of Alif one points, and it was sort of the beginning of this this thing that that is still my dream that there might be the different levels of infinity would correspond to different physical.

Ken Taylor
Here’s another xenon paradox, though. Right, take a point, take an extensionless point, and put a whole bunch of extensionless points together. What do you get? No extension because zero plus zero equals zero. Yes, right. And obviously there is extension, but it’s not because extension is made up of an infinity of points. An infinity of points gives you no extension. So extension has to be something over and beyond the right.

Rudy Rucker
It’s a tough paradox. It really is, and maybe if you have aleph null points, they do add up. Or you you could go into the serial number mode and say, okay, we actually have absolutely infinitely many points.

Ken Taylor
But but I want my point was trying to say the infinite divisibility of the number line doesn’t entail anything about the infinite divisibility of space because the infinite points doesn’t give you extension. Extension is something else. Yes. We don’t quite know what that something else is, but it’s something else.

John Perry
You said a little Aristotelian here. You really want to just keep the potential infinite.

Ken Taylor
Oh well. Well, there is this

Rudy Rucker
thing of we do have the mathematical notion of infinitely many points in space, but whether that matches what’s going down in the very small regions of physical space is something that’s really up for grabs.

Ken Taylor
Yeah, and I guess we’re still grappling with. I mean, it’s worries about this that led the early Greek atomists, right, to say there must be a atom that can’t be divisible divided any further.

Rudy Rucker
Yeah, atom means no cut. Right. So there you go.

John Perry
That that definition didn’t turn out to stick with Adam, though. Well, that’s it. They keep

Rudy Rucker
finding new stuff. I mean, you’ve got the quarks, you’ve got the gluons. They really

John Perry
are sleaze bags. They can’t even stick with the Greek definition of their basic term. But

Ken Taylor
Silver Rudy, I’ll give you chance for one last comments, maybe about experience of infinity. Since you’ve tell me one last thing about the experience of infinity.

Rudy Rucker
Well, it’s I like this idea of getting into my head and going out into into the infinite. It’s just recently I was I’ve been writing a story about somebody that leaves this body and goes out beyond the the furthest galaxy, and then can do things like count infinitely many numbers on a tree, and I like getting into these worlds.

Ken Taylor
Well, on that note, I’m gonna have to thank you very much for joining us.

Rudy Rucker
Thank you. It’s been really interesting.

Ken Taylor
Our guest has been Rudy Rucker, professor emeritus of computer science at San Jose State University, author of many things, including Infinity of the Mind, but also many novels and other accessible books. So, John, what did you learn today?

John Perry
What did I learn today? Well, I I kind of was reminded of a number of of things I’ve learned or almost learned in the past, and it kind of came to the same conclusion. On the one hand, our last our last emailer, Fred Eday, seemed to me to have a really compelling point. I mean, if it’s hard for me to imagine that that space isn’t infinitely divisible, and if so, if a physical object takes up some space in a way we’re we’ve got a completed infinity in front of us. Well, right. On the other hand, all the arguments that Cantor gives us are always impressive and and and you know airtight, but they always do have a little air of hocus pocus about them.

Ken Taylor
What’d you learn? Well, I mean, I was fascinated by lots of things that Rudy said. I I learned a lot, you know, and I learned that you could be a rigorous, hard-thinking, mathematically sophisticated, computationally sophisticated guy, and also be you know kind of a mush. Transcendentalist kind of thing that these two things don’t necessarily that they actually can complement each other. I thought that was actually fascinating the idea that you know because you think of the religious traditions and thinking about infinity, and you think that’s kind of mystical, right? And all about experience, but then you think well, the mathematical thinking about infinity works out some of the potential content of that mystical experience, and they maybe they can be put together. So, so

John Perry
are are you on the verge of being the first person converted to religion by set theory?

Ken Taylor
No, well, I think no, I would I wouldn’t say that. It would it would take a little bit more for me to do that. But you know what? This conversation continues on our blog, the blog dot philosophy talk.org, where our motto is cogito ergo blog. Now our blog has been is been lacking a bit in recent days, but we’re gonna get back to that thing. I I promise you, we’ll get back to that.

John Perry
I was gonna blog on infinity, but then I got an infinitely bad cold, and I’ve just been sleeping most of the time since then. But you know, for the final word on infinity, can you have a final word on infinity? Well, at any rate, for the final word on infinity, we turn to the fast, although not infinitely fast-talking Ian Sholes, the Sixty-Second philosopher.

Ian Shoales
Ian Sholes, nowadays infinity is pretty much taken for granted, at least in the world of mathematics. Good old-fashioned wonder, staring up at the stars is still around, of course, but most of the time, infinity is an excuse to invent thought experiments, paradoxes, riddles, and metaphors. The one thought experiment that puzzled me the most is the one that posits that if you had a monkey and a typewriter with an infinite amount of time, the monkey would eventually write Hamlet. Well, first of all, why a monkey? Why not a duck or even a human being? It would probably be less likely that an infinite human would write Hamlet because he would be trying to write Hamlet, based on having read the Cliff Notes in high school, in infinite monkey world, I would suggest it would be easier to write Hamlet by accident than intent. Still, what is it with monkeys? It’s not just in thought experiments. Monkeys show up everywhere. There’s monkey as moral arbiter. Monkey see, monkey do. They hear no evil, see no evil, speak no evil monkeys. On the other hand, sex is sometimes called monkeying around. If you’re an addict, you have a monkey on your back. A barrel of monkeys is that against which all fun is measured. The brass monkey and body part of same is that against which great cold is measured. Mischief is monkey shines. When you go to stop something in its tracks, what do you throw into it? A monkey wrench. And we have monkey business, sock monkeys, monkeys, uncle, monkey bars, cheeky monkeys, sea monkeys, grease monkeys, code monkeys monkey business. The signifying monkey. Unless we forget, if you peep peanuts, you get monkeys. Don’t make a monkey out of me. And there’s this funny monkey, Dr. Lyle Watson, in his 1979 book Life Tide wrote, “An unspecified number of monkeys on Koshima were washing sweet potatoes in the sea. Let us say, for argument’s sake, that the number was 99. At 11 o’clock on a Tuesday morning, one further convert was added to the fold in the usual way, but the addition of the 100th monkey apparently carried the number across some sort of threshold, pushing it through a kind of critical mass, because by that evening almost everyone was doing it. This led to the 100th monkey theory that there’s some kind of neo telepathic tipping point in societies after which behavior is to catch on. The 100th monkey is responsible for everything from fads to social mores to the death of the VCR. So why just one infinite monkey at a typewriter? Why not have 100 infinite monkeys? If the 100th monkey theory holds, when one of them writes Hamlet, eventually all of them will. Either that, or we’ll have 100 smashed typewriters. With that much time on your hands, breaking stuff is a lot more fun than writing. Believe me, I’ve been there. I gotta go.

Ken Taylor
Ian shows the only man who can solve a philosophical problem in 60 seconds.

John Perry
Philosophy Talk is a presentation of Ben Manilla Productions and the trustees of Leland Stanford Junior University, copyright 2008.

Ken Taylor
Our executive producer is David Demarest.

John Perry
Our production coordinator is Devon Strolovitch. Daniel Elstein is our director of research. Lael Weiss is our webmaster. Also, thanks to Zoe Cornelli, Merle Kessler, Corey Goldman, and Mark Stone.

Ken Taylor
Philosophy Talk is sponsored in part by Powell City of Books. On the web at powws.com. Support also comes from the Templeton Foundation

John Perry
and from various groups at Stanford University, the Friends of Philosophy Talk, and the members of KALW San Francisco, where our program originates.

Ken Taylor
The views expressed or misexpressed in this program do not necessarily represent the opinions of Stanford University or of our other funders.

John Perry
The conversation continues on our website, philosophytalk.org. I’m John Perry,

Ken Taylor
and I’m Ken Taylor. Thank you for listening,

John Perry
and thank you for thinking.