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.