Today, we're going to talk about the question...
"How do we know about numbers?"
When we ask...
How do we know about people, or trees, or any stuff like this...
It's kind of obvious.
You go out there and you look at it...
... or you go and ask an expert who goes and looks at it, and the expert tells you about it.
But numbers...
... or other mathematical entities, like sets or functions....
They're not...
...out there like that.
You can't go and look at them.
You can't listen to them.
You can't go and see a film about them.
So...
That's how we get stuck with this question.
How do we interact with them?
How can we kind of...
...get our hands on numbers...
... in a way that gives us information about them.
The way you find out about the world...
...in nearly every case...
...is by having some physical, causal interaction.
Either you see it
you hear it
you smell it
...or someone who's done those things tells you about it.
That's all a causal interaction.
But numbers
At least on most people's theory
They're not physical things.
They don't causally interact with us.
So that raises the question:
Brady "Do you know the answer to the question?"
Mark "I guess I don't KNOW the answer
but let me tell you about some answers people have come up with.
So here's one answer:
Numbers aren't out there in the world
they're really stuff we make up in our heads.
The idea is numbers are these things we construct in our own minds
It's a special thing about human minds
that allows us to construct numbers and other bits to do with maths.
If that theory works,
it's got a really nice answer to the question "how do we know about numbers?"
well, it's because we've just made them up in our minds.
We know about the stuff in our own minds, numbers are in our minds, so that's how we know about numbers.
That goes by the name Platonism.
Plato thought that properties like redness, and justice, and goodness
exist but not in the physical world
they exist in a world of forms.
Platonism about mathematics is kind of like that,
it's saying that these things really exist, but they are non-physical things.
So you might think that fictional entities, like Sherlock Holmes,
Or mythical entities, like Pegasus, they exist in our imaginations in our minds,
in the minds of authors, and then they get passed on to the minds of readers.
So that analogy, between on the one hand fictional entities, like Sherlock Holmes,
and on the other hand, numbers-
Brady "i.e. seven?"
-yeah, being creations in the mind.
That gives you some clue to why this theory about numbers has problems,
because on the one hand
it's really obvious to think one day somebody came along and invented Sherlock Holmes
It doesn't seem as if somebody came along and invented the number seven
there's always been the number seven
at least that's what I think, there's always been seven things
so how could there have not been the number seven?
So imagine we rewind the clock way back to the days of dinosaurs
suppose the dinosaurs aren't that clever, they don't have any conscious experiences,
if you've got 24 dinosaurs roaming around
it seems as if there's 24 of something
so you've gotta have the number 24.
So if we're not there to be thinking about it,
it seems as if you can have numbers without human minds thinking about those numbers.
Brady "you don't think that numbers are a human-made tool, you think that they're beyond that?"
I think if this theory was to run, it'd have to be made a bit cleverer.
There's a way to do that.
And that's to think of numbers as not as things that we actually create in our minds
but rather you think about a number as
the potential for us to do something.
The idea is something like this.
Even before humans came along, there was always the potential for us to start counting.
Going, ah, there's three trees, there's 24 dinosaurs.
If that potential's there, then the numbers are there.
That's how that theory goes.
Here's a completely different way of thinking about the matter.
What do we do when we do maths?
Well, we look at some equations, we go through them, we try and solve them,
we look at some axioms, we try and prove some stuff, we talk to mathematicians,
and they do some squiggles on the boards...
So here's an idea. Mathematics, numbers, really just is about those equations, those axioms.
So on this view, there really isn't anything to numbers beyond the kind of symbolism we have for them.
This kind of theory has got the same kind of problems that I just talked about.
Obviously there was a time before we had any kind of symbolism, you know, before we were around.
So, how would you then explain numerical facts about the past?
So these guys, if they're gonna try and make this theory work, they have to somehow explain,
when we describe the past, those 24 dinosaurs, we're kind of using symbolism to do it.
We'd write down the number 24, or we'd make the sound 'twenty-four'.
So, even when we're describing the past, we're describing it now using these symbols, and that
kind of gives some support to the idea that the number really doesn't go beyond the symbol we use for it.
And that goes by the name either intuitionism, or constructivism.
That's actually really big amongst computer scientists
who think of numbers not so much as constructions in our mind, but constructions in a computer.
I think we're in a bit of a bind when we come to thinking about our knowledge of numbers,
because I think numbers are real things, I think they're objective things, so that
when I talk about the number 23, and when you talk about the number 23, we're talking about the same thing.
I don't think the number 23 is something that we could bump into in the street. I think that it exists,
but it doesn't have a physical presence in the world.
Numbers don't bash into each other like, you know, cars or snooker balls do.
So the bind is something like, numbers are real non-physical things,
on the other hand, we know about the world by physically interacting with it,
that looks a bit like saying we don't know about numbers,
but I think we do.
So here's one suggestion about how we solve this problem, and it goes like this.
Numbers are real things, they're out there in the world, they're non-physical things.
How do we know about them?
We say that if it could possibly exist, it does exist.
Suppose we come up with some mathematical theory.
We go through all the axioms, we check all the proofs, and we go, ah yeah, this is consistent, this looks good.
So these numbers, or these things that our theory talks about, they could exist.
And then the philosophy goes, since it could exist, it does exist.
So we know that because our theory is consistent, the stuff it talks about really does exist.
Brady "I can imagine a lot of people who watch numberphile aren't gonna like that...
just because something can exist it does exist?"
Right. And in general, that's just crazy, right? I mean, Sherlock Holmes could exist,
doesn't mean he does exist.
But you might think that mathematical entities, because they're not physical, they don't take up any space or time,
are different.
So the mathematical universe seems to contain all these weird and wonderful things,
and it almost looks sometimes like we can define things into existence.
We can say things like, let's define a group out of sets as blah-blah-blah
Now how would that work? How could you just define something into existence?
This theory has an answer. You're not making it exist.
What that definintion is doing is saying,
That goes by the name formalism.
The formalism is the formal system of symbols that we write down,
the axioms, the proofs, the rules we set ourselves
to say that's a good proof, that's a bad proof. Once we've got those symbols, and we've got those rules,
hey presto, you've got the numbers.
Interestingly, most of these questions were originally raised by mathematicians.
And they were raised in a time, back in the early 20th century, where lots of areas of maths had gone bad.
They discovered that lots of the theories they were using gave really weird and contradictory results.
It was paradoxes discovered in the foundations of mathematics.
So one really important concept was the concept of a set.
This is an idea that came from Cantor, and Cantor thought, well, a set is defined by a predicate,
like, "is tall". "Is tall" defines the set of tall people.
And the idea was, you give me a predicate, I'll give you a set.
But Bertrand Russell, in his famous paradox, showed why that's not gonna work.
He said, take the set of all things that aren't members of themselves.
That's a predicate.
If you try and make a set out of it, you get a contradiction.
So, established area of mathematics, we thought we knew it, gives us contradictions.
And the whole industry based around making sure those paradoxes went away
really was a search for an answer to the question, "how do we really know about mathematics?"
What the fictionalist has got to do is try to explain away the success of mathematics and
the success of science, but without saying that mathematical claims are true.
The mathematical theory, on the other hand, tells you, bet whatever you've got,
stake any amount of money you can get your hands on.