Welcome back to The Very Short Introductions Podcast. From
public health to Buddhist ethics, soft matter to classics,
and art history to globalization, we'll showcase a
concise and original introduction to a wide range of
subjects for, wherever your curiosity may take you. So here
is today's very short introduction.
My name is Ian Stewart. I'm a retired professor of Mathematics
at Warwick University. And being a mathematician, I've run into
infinity in some form or another, virtually every day of
my mathematical life. And it's a fascinating topic. It's not just
mathematicians who are interested in the infinite.
Philosophers, naturally, are fascinated by the rather
complicated and paradoxical nature of infinity. And
theologians have debated the notion of infinity relative to
various deities in various religions. I want to mostly
focus on the maths.
But we run into infinity very early on. If you ask children,
what's the biggest number; when they're very young, they will
name the biggest number they know. So, depending on the age,
they'll say, 10, or 100, or a million. But, fairly early on,
they realize that however bigger number they name, there's always
a bigger one, and the easy way to get a bigger one is just to
add 1. So if you think of two children playing a game, see who
can name the biggest number, and keep going until one of them
can't. Well, basically, the game goes on forever, because very
rapidly, they say, "Oh, one bigger than what you just said."
And they keep doing that. So there is no biggest number.
So in some sense, the collection of all possible numbers--I'm
just thinking of whole numbers here--is infinite. And we kind
of get used to that, we don't think very much about it. But if
you start examining it more closely, it's a very tricky
notion. And in fact, there are many different kinds of
infinity, as I'll get around to.
So the first recorded investigations of infinity are
by ancient Greek philosophers. About 580 BC, Anaximander,
wondering where everything in the universe came from, came up
with something he called "apeiron," which is a kind of
formless chaos, out of which everything is created. It's
infinite, so you can make things from it without ever running
out. It's a bit like the quantum mechanical idea that particles
and antiparticles can suddenly wink into existence from a
vacuum. But I don't think the ancient Greeks were really
thinking about quantum mechanics at that point. So it's a
slightly vague concept, but it's the first recorded one.
The first mathematical analysis of the infinite is about a
hundred years later. And this is the ancient Greek Zeno; you may
well have heard of Zeno's paradoxes. And Zeno came up with
four different paradoxes-- paradox meaning that the logic
just did not work properly, it didn't matter whether you
thought something was true or false, you ran into trouble. And
the most famous one, and the one I'm going to talk about, is
Achilles and the tortoise. So Achilles, who is big and hulky
and very, very athletic, he's going to run a race against the
tortoise. And he knows the tortoise is a lot slower. So he
said, "Okay, look, I'll give you an advantage. You can start 10
meters further ahead than me, and then we'll start the race.
And I'm going to beat you." And the tortoise said, "Oh, no, you
won't, by the time you get 10 meters, I'll have moved on a
bit. And by the time we get to where I was, then I'll have
moved on a little bit more. And so you will have to do
infinitely many different things before you can catch me." Of
course, they run the race, and Achilles runs past the tortoise
at high speed. And the tortoise is sitting there thinking,
"Well, the logic seemed to work, but I've lost the race." So this
is an example of what kind of assumptions you make about
infinity. Is it actually possible to do infinitely many
things in a finite time? And the modern answer is that in the
mathematical models we use, yes, it is. Basically Achilles is
right, the tortoise is wrong. But it's not always that simple.
Aristotle, who is perhaps the most famous of the Greek
philosophers, came up with a very important distinction
between what he called actual infinity and potential infinity.
So let's go back to the whole numbers. You start counting 1,
2, 3, 4, 5. And this process can go on and on, and, in principle,
it never stops, because, however far you get, you can always go a
bit further. Now, that's an example of what Aristotle called
potential infinity. We never actually get to infinity, we
just keep going and never stop. But at any time, we've only
counted finitely many numbers. But there isn't a biggest one.
So potentially, the number system is infinite, but we never
grasp the entire infinitude of it as a single thing. Actual
infinity would be something that is of itself genuinely infinite.
And actually, most uses of infinity in mathematics are
potentially infinity or can be recast and reworded so that
you're never actually facing actual infinities. And that was
true until about 1850 when the whole game changed.
Throughout history, people, often very famous people, have
come across infinity, thought about it, noted various
interesting properties of it, and given their view of what's
involved. And Galileo, the great Italian scientist and
mathematician, wrote a book in 1638, Discourses and
Mathematical Demonstrations Relating to Two New Sciences.
And he had some characters who were discussing things. And at
some point, they're discussing numbers, and one of them says,
"Well, we all know about the numbers, 1, 2, 3, 4, 5, 6, and
so on. And we know what the square of a number is; to square
a number you multiply by itself, so 1 squared is 1, 2 squared is
2 times 2, that's 4, 3 squared is 3 times 3, that's 9. Sixteen,
25, 36, and so on. And a lot of numbers are not squares. So
rather, obviously, there are more numbers than there are
square numbers." The other one says, "Yes, but I can match them
up. Every number corresponds to a square number, namely, its own
square. So the first square is 1, the second square is 4, or
the third square is 9, and so on. And since I can match up
every number with every square, and conversely, there are
exactly as many squares as there are numbers." So on the one
hand, there are obviously more numbers than squares. And on the
other hand, the numbers are obviously equal. And Galileo,
basically, at that point, throws up his hands and says, "Well,
what this shows is that talking about equality, greater or less,
doesn't apply to infinity." And for a long, long time, it's left
like that.
Now the other place that we encounter infinity is in
arithmetic, recurring decimals. We learned about fractions, we
learn about decimals, we come across 1/3 and we're told it's
naught point 33333 going on forever. And that the number pi,
which comes up in connection with circles, is 3.141, etc.,
and that goes on forever, as well. And one of them, it keeps
repeating the same number and the other one doesn't, but they
both go on forever. And at school, we kind of slide over
all of that. The teachers don't want to get too involved in what
it means. But mathematically, this is a very subtle point,
that decimal numbers that go on forever are not something you
can actually write down. You can write down a rule to calculate
them, but you can't write them down. And that's a kind of
potential infinity as well. We can always work out the first
hundred decimal places or the first thousand but we can't
write down the lot. Why is this important? Why should we care?
Because a lot of today's society actually depends upon the
mathematics of infinity, and have the kind of flipside of
infinity which is the infinitely small, infinitesimals, numbers
or things like numbers, which are so small, that in certain
circumstances, you can almost pretend they don't exist. And
this is the basis of calculus.
So calculus is used to find areas, to find volumes, to
calculate the speed of objects, to calculate accelerations, and
the whole of mathematics of calculus rests upon adding up
infinitely many infinitely small pieces and getting a sensible
answer. And so the mathematicians of the 17th,
18th, 19th centuries had to grapple with that idea. The
engineers and scientists and the physicists and chemists just
picked it up and ran with it and said, "We don't really care
about the foundational details of this, this stuff works, let's
use it." And this is why we now have satellites in orbit. It's
how it's possible to land on the moon. It's how cars are
designed, it's kind of how bits of the Internet work. It's how
electricity works. All of these things we understand using the
mathematics of calculus, which is founded upon notions of
infinitely large and infinitely small. So this stuff is actually
useful, it's indispensable.
A very different place where we encounter infinity is,
surprisingly, in art, in paintings. Before the
Renaissance period, paintings were often very beautiful, but
slightly crude in the sense that things like perspective of
buildings and landscapes didn't really work, that the painting
didn't look like a photograph. It was a nice painting. The
Renaissance painters discovered mathematical techniques for
getting very, very accurate perspective. And so when they
draw a landscape with buildings, and maybe it's Venice, and
there's a canal, the stuff up front looks bigger, the stuff at
the back looks smaller, but also the whole thing fits together so
that the eye sees it almost as a photograph of a three
dimensional object. And infinity comes into the mathematics of
that; think about railway lines, suppose you are standing--it's
bit dangerous--but you stand in the middle of a straight railway
line, which is going on for some considerable distance. And
basically, when you look at it, the lines seem to converge,
because the further away the lines get, the closer together,
they become, as far as the eye is concerned. They're actually
parallel, but they seem to get closer together. And if you do
this using Euclid's geometry, then you would draw parallel
lines on a plane and you would stand between them and look to
where they go, and they would seem to converge to a point that
is infinitely far away. And it's an entire line like a horizon,
which looks like the edge of the infinite plane. But an infinite
plane doesn't have an edge. So there is something called the
line at infinity, which is a geometric example of a concept
motivated by thinking about the infinite.
And it leads to what's called projective geometry. In Euclid's
geometry, two straight lines either meet or they're parallel
and they don't meet. In projective geometry, two
straight lines always meet; the parallel ones in Euclid's
geometry meet on the line at infinity. So you're kind of
completing the infinite plane with a tangible boundary, where
lines can meet. And so projective geometry became very,
very important, both for art and later for mathematics.
What about the real world? Is anything in the real world
genuinely infinite? While not too long ago, I think most
physicists--and they're the people who grapple with this
mainly--would have said, mostly not. Usually, when you meet an
infinity in your calculations, it means your theory is breaking
down. There's a theory of the rainbow to do with how the light
is focused by a raindrop. And the classical version of this
predicts that the intensity of light on the rainbow should be
infinite. And clearly it's not. So you need a slightly different
theory there. It's a wave theory of light, and then things work
out better. But one question is, how big is the universe? And
basically, we have no idea. It's very, very big. There's a limit
to what we can observe. And physicists do discuss whether
the universe is actually infinite, and we're just seeing
a finite portion of it, or whether the whole thing is
finite; it might even be curled up into a sort of
four-dimensional ball. So there's a possibility there for
a real infinity.
The other place where you might find a real infinity, certainly
in the theory, is a black hole. And you will have heard of a
black hole; it's what happens when a very heavy star collapses
under its own gravity. And it collapses so much and become so
dense, that the attractive force of gravity stops light escaping
from it; the light can't get fast enough to escape the
gravitational field. Now, the theory is that inside the black
hole, all of the matter that collapsed actually condenses
right down to a single mathematical point of infinite
density. We can't get inside a black hole so we don't know what
the reality is, but at the moment, the best theories we
have are predicting a real infinite density point. My
suspicion is that that will eventually be overturned by a
bit of theory, but that's a case where real infinities are
turning up in physics.
And the final mathematical point I'd like to make was the amazing
discovery by Georg Cantor, in 1874, that some infinities are
bigger than others. Now, that sounds ridiculous, it's
infinitely big, how can anything else be, some infinite because
we even bigger than other infinitie? And his example was
the counting numbers, 1, 2, 3, 4, 5, and so on, going on
forever, thought of as an actually infinite set, this
isn't actual infinity not potential anymore. That's a
smaller infinity than the infinite decimals, all of the
possible infinite decimals, .33333, recurring, pi, 0.714526,
you name it, it's up to you, carry on forever. Cantor proved
that there are more infinite decimals than there are counting
numbers. You can't match them up like the counting numbers match
Galileo's squares. So there are different sizes of infinity.
And so I'd like to leave you with the thought that infinity
starts out as a sort of philosophical concept, is turned
into mathematics, the mathematics is turned into
physics, and then some of the mathematics comes up with things
that nobody had thought of, until a couple of 100 years ago,
which also, in fact, have turned out to be very important. So
infinity is beautiful, puzzling, and well worth thinking about.
Thank you for listening to The Very Short Introductions
Podcast. You can subscribe to our podcast on your favorite
app, such as Apple or Spotify, to receive all of our episodes
directly in your feed. All of our episodes, new and old, can
also be found on SoundCloud and YouTube at Oxford Academic.