Uploaded by numberphile on 03.06.2012

Transcript:

DAVID HODGE: So what are we going to be talking about

today is whether or not numbers exist.

We're going to be thinking about three different schools

of thought on this.

First one's Platonism, second one's nominalism, and the last

one is fictionalism.

So mathematical Platonism takes its lead from Plato,

obviously, as the name suggests.

And the idea is that numbers exist.

They really are things.

They're objects.

But they're abstract objects, so they don't exist in the

same way that tables and chairs do in

space and in time.

Numbers exist outside space, outside time.

They don't cause other things to happen.

They're totally abstract objects.

One of the reasons that people like mathematical Platonism is

that we think that a lot of claims about numbers in

mathematics are true.

Right, so certainly I think that things that

mathematicians say are true.

I think that's a good starting point.

But they say things like there's a number

between 6 and 8.

That seems like it's a true thing, it's the number 7.

And Platonists think that that means that there must exist an

object that is the number 7, even if it's not an object

that you can touch.

And they appeal to kind of similar reasoning about the

way that we speak about the ordinary,

kind of concrete world.

So if I say to you that there's a nose between two

eyes, that nose had better exist for what I say to be

true, right?

If there's no nose here, then what I've said is false.

So in the same kind of way, right, if there's a number

between six and eight, that thing had better exist.

Platonists aren't happy to use a loose sense of "exist" here,

so they think that exists just means exists.

And so in the same way that a table exists, a chair exists,

a number exists.

All the numbers exist, in fact.

It just so happens that they're not located

in space and time.

So it's a difference about location, rather than whether

or not they exist, or whether they exist in a

different kind of way.

All sorts of fun problems with the view.

The first of which that you come across, particularly when

you start teaching this stuff, is that's crazy.

So numbers exist outside space and time?

What does that even mean, right?

It's really, really hard to kind of get your head around.

But we can be a bit more refined than that when we're

objecting to mathematical Platonism.

So one of the things that we said was that mathematical

claims are true, but that mathematical claims are

actually claims about these abstract objects.

These abstract objects don't causally interact with us

because they're abstract.

OK, but we think that mathematicians

are massively reliable.

So what they say, pretty much guaranteed to be true.

If their claims are actually about these abstract objects,

then it looks like we need some explanation of how those

mathematicians so reliably track that

world of abstract objects.

So if a mathematician tells you that 6 times 7 is 42, then

what he's saying is true.

What they're saying is true.

How is it that they're so reliably managing to access

this world of abstract objects?

Answering that question's really hard for the

Platonists, because they just don't think we can interact

with that world.

Our second theory is mathematical nominalism.

So what the nominalist wants to do is

agree with the Platonist.

That all of our mathematical claims are true, but that our

mathematical claims should best be understood as claims

about objects in the world.

So tables, chairs, pencils, those kinds of things.

And to get a kind of bit of intuition on this kind of

thing, often helps to think about the way you might teach

a child to count.

So children don't tend to be born with these kind of

abstract principles.

One, two, three, four, five, six, and so on.

What we tend to do is to say, like, here's a pencil.

Here's another.

Now you've got two pencils.

And it seems that what we do when we learn to count is we

count with respect to a particular type of objects, or

to objects in general.

And what the nominalist says is, well, that's all there is

to number talk, right?

What you mean when you say 6 times 7 is 42 is that if you

had 6 lots of 7 objects, you'd have 42 objects.

That's what mathematics is about.

It's about concrete things in the world.

This kind of nominalist view begins to run into trouble

when we start to think about more complicated numbers or

numerical concepts.

So if we start to think about the square root of minus 1,

for instance, and the imaginary numbers and so on.

So it's really easy to say 2 plus 2 equals 4 is about these

2 objects and those 2 objects, and there being 4 objects.

Much harder to see what object or thing might be the square

root of minus 1.

So if you're a mathematical Platonist, it's really easy.

It's just another number, just like all the rest.

But if you're a nominalist, what's the

thing that it's about?

Looks really, really hard to see what that's going to be.

So pi is also similarly complex, right?

So if you're a nominalist, it looks like you're going to

think that numerical talk is just talk

about concrete things.

But of course, pi doesn't seem to have a precise value, or at

least we certainly can't calculate that value.

So there's a real question for the nominalist here.

How are you going to understand pi as a thing, or a

collection of things, when we can't ever get to that

terminating value?

Our third group in the philosophy of mathematics is

the mathematical fictionalist.

So I said at the start that I think it's a good thing to say

that what mathematicians say is true.

And the fictionalist doesn't quite agree.

So the fictionalist says, actually, mathematical

discourse is false.

It's really useful, really helpful, but it's

systematically just false.

That's quite an extreme view to take, given the success of

science within which mathematics is embedded, but

that's the view that the mathematical

fictionalist takes.

So the fictionalist says numbers don't exist, whereas

the Platonist says numbers are these existing abstract

objects, and the nominalist says mathematical discourse is

about concrete objects in the world.

The fictionalist says it's not about anything.

It's just a useful story that we've developed to help

ourselves get on in the world.

BRADY HARAN: But surely the mathematical fictionalist uses

modern communications, is dependent on satellites.

How do they explain those things?

Because obviously they use numbers.

How do they explain that they're using something that

they say doesn't exist?

DAVID HODGE: So what the fictionalist has got to do is

try and explain away the success of mathematics and the

success of science.

But without saying that mathematical claims are true.

And one strategy that the fictionalist can deploy around

here is to say that success in the world isn't a hallmark of

truth, or needn't be a hallmark of truth.

We were talking about modern communications, technology

relying on all of this mathematics.

What the fictionalist will say is what that indicates is that

the mathematics is successful.

It doesn't indicate that it's true.

If it was true, says the fictionalist, these

mathematical objects, these platonic objects that the

mathematical Platonist thinks exist, those things

would have to exist.

But they don't.

And so in fact, our mathematical

discourse can't be true.

It can at best be a useful fiction.

There are appeals that they can make around here that kind

of help bring out the sort of thing that they

might want to do.

Imagine that you thought that the Bible was a work of

fiction, for instance.

Some people do, some people don't.

But you might think it codifies a really useful set

of moral principles.

And you might think that there's no such thing as truth

and morality.

But you might think as guiding principles go, these are

pretty helpful.

They help us get on in the world.

Don't kill other people.

Help societies develop, which helps us develop in

interesting kinds of ways.

The mathematical fictionalist might say very similar things

about the language of mathematics.

Helps us get on, but that it helps us to get on doesn't

mean that it's true.

BRADY HARAN: Where do you stand on all this?

You've very neutrally given me the three positions.

Is there one that resonates with you?

DAVID HODGE: So the view that resonates with me the most, I

guess, is mathematical nominalism.

So I don't think that there are abstract objects.

I don't think that there are numbers.

I think that it's too much of a stretch to think that

mathematics is false.

My colleagues in the math department will, I'm sure, be

delighted by this.

So I adopt the middle ground, what I take to be the middle

ground, anyway, which is a kind of mathematical

nominalism.

BRADY HARAN: You were telling me that the nominalist runs

into problems with some of these more spectacular

irrational numbers.

Do you run into problem with those numbers?

DAVID HODGE: Yes, that means that I inherit all of the

problems that the mathematical nominalist faces.

The kind of move that I like to make around here with

respect to pi is to say that whenever we actually calculate

with it, when we're using it mechanically, if you like,

what we're doing is we're using a best approximation.

Now, that's a spectacularly accurate, well, it's a

spectacularly long approximation, anyway.

I really shouldn't say it's accurate, because I don't

think it maps to a platonic ideal.

But it's a spectacularly long and very useful approximation.

But I don't think there's anything other than that

endless sequence, I guess.

And of course, you can derive that perfectly acceptably.

I think the reason that philosophers get into the

philosophy of mathematics, one of the reasons, anyway, is

precisely because there are so many things to

philosophize about.

It's another thing for us to get excited about and

interested in.

The grand tradition of thinking about numbers, it

goes right back to Plato, hence the

mathematical Platonism.

And so people just get excited and interested in it, I think.