I'm going to talk about, give a general
introduction to, the philosophy of mathematics. And I'm not going to assume
any knowledge at all of either philosophy or mathematics. So I'm sorry
if it seems a bit basic to some of you. Okay. I mean, the first thing to say, I
think, is that mathematics and philosophy which I think most people tend to think
are poles apart, are actually closer than you might think. And a lot of the great
philosophers have also been great mathematicians. And I think that's not
an accident. I think there's a close affinity actually between the two subjects in
several respects, the most noticeable perhaps is that both require and
demand thinking on a very abstract level. But as I hope to show today, there's a lot
more to the relationship between the two than that, and in particular, that
mathematics has furnished philosophy both with a model of a certain kind of
knowledge and with a set of deep and interesting philosophical problems.
But the two go together. Plato knew this. It's said that at
the opening of Plato's, at the gate of Plato's Academy, it said: "let no one
into here who knows no geometry." Interesting, I think, that it's
geometry rather than arithmetic. And there's a story there which I'm
going to tell which is to do with the fact that the ancient Greeks regarded
geometry rather than arithmetic as the more foundational, the superior
branch of mathematics. And one of the main reasons for that was the unhappy
story of Pythagoras. Most of you I think will know the name Pythagoras from his
famous theorem which we all learned at school which says that the square on
the hypotenuse of a right angle triangle is equal to the sum of the
squares on the other two sides. Well, that very theorem presented a
problem to Pythagoras and his followers. I don't know whether many of you know this
but Pythagoras was a sort of cult figure in ancient Greece, and he had a band
of followers who were dedicated to the mysticism of numbers. It was one of their precepts that
everything in the world could be expressed as number, and in particular,
as the ratio between two whole numbers. Well now, if you go back
to Pythagoras's theorem, it's a consequence of that theorem
actually that that belief is unsustainable. And the reason for that is
that reflecting upon that theorem leads you straight into what are called
irrational numbers, numbers that cannot be expressed as the ratio between two
whole numbers. To see this, imagine a right angle triangle with length 1 here
and length 1 here. So you've got a square on here of 1, a square on here of 1.
The square on the hypotenuse therefore is going to be the square root of 2.
And Pythagoras discovered to his horror that the square root of 2 cannot be
expressed as a ratio of two numbers. It's not a rational number. Pythagoras was so horrified by this he
swore his followers to secrecy about it. They weren't allowed to mention
the irrationality of the square root of 2. But it did undermine the faith in
numbers as the foundational view of mathematics. And that's why geometry is
regarded as superior because in geometry you can express the hypotenuse,
you can draw it. But in numbers, you can't express it as a ratio. Okay. So,
mathematics and philosophy have always gone hand-in-hand. I said one reason
for that is that mathematics provides a model of knowledge. It provides a
model of knowledge of a particular kind. That's to say, philosophers since the
ancient Greeks, for over 2,000 years, have regarded mathematical knowledge as
somehow special and something to which all knowledge could possibly aspire. What
makes mathematical knowledge special? Well, several things. One is, unlike other knowledge, it's certain.
If something is true in mathematics and if you know it, you're not going to doubt it.
You're not going to say, "well, I'm fairly sure that 2 + 3 = 5", you know that 2 + 3 = 5
with absolute certainty. That's the first thing. The second thing is
that knowledge in mathematics seems to be incorrigible, that's to say, it's not
going to be corrected by anything you might subsequently learn. 2 + 3 = 5
now, 2 + 3 = 5 two thousand years ago, and you might say, even if
there weren't any people on earth, it would still be the case that if 2 apples
fell to the ground and 3 more apples fell to the ground, then you would have
5 apples on the ground altogether. So mathematical knowledge is certain, it's incorrigible. Another thing is, that's
related to its incorrigibility, it's eternal. It's always true. If something is
true in mathematics, it's not true for the time being. It's always true and
always has been true. And a fourth thing about mathematical knowledge is that it
seems that mathematical truths seem to be not just contingently true,
but necessarily true. Southampton happens to be on the coast
of England. But that's not a necessary truth, it doesn't have to be. I happen to be wearing black trousers.
But again, that's not necessary. But 2 + 3 could not equal anything
but 5, it's necessarily equal to 5. So, philosophers have looked at this example
of mathematical knowledge and they've thought two things, one is: How is it that
knowledge in mathematics has those characteristics? What is it about
mathematics that gives its knowledge those characteristics? And the
second thing that's occurred to several philosophers including Plato, Hobbes,
Russell was: Why can't other kinds of knowledge be like that?
Maybe we could have a system of physics, for example,
that made our knowledge of physics as incorrigible, eternal, certain & so on
as our knowledge of mathematics. So, mathematics has provided a
model of knowledge. On the other hand, it's inherently puzzling. Mathematics
has provided philosophers with a number of deep puzzles that we have been
scratching our heads over for over 2,000 years. The central one of which, is the
most basic of which, which is: What is mathematics about? So, if
you know that 2 + 3 =5, you know with certainty, incorrigibility, and so on. But
what is it that you know something about? As I said if you add 3 apples to 2
apples, you'll get 5 apples. If you add 3 lemons to 2 lemons, you'll
get 5 lemons. But 2 + 3 = 5 is not about apples, and it's not about
lemons. It's applicable to those things, but it's not about those things. So, what is it about? Well, it's about
numbers. But what are numbers? It's when you ask that question–it's one of those
philosophical questions that, the more you think about it, the less clear it gets.
What are numbers? Numbers seem to be the content of objective truths. If I say
that 2 + 3 = 5, I haven't just made it up, I'm not getting it from anybody. It is
objectively true. And most of the things that we have objective truths of
are objects. So are numbers objects? Well, unlike apples and lemons, you can't
see a number, a number does not reflect light. You can't smell a number, you can't
touch a number. So if a number is an object it's a peculiar kind of object.
It doesn't exist in space and time, it doesn't corrode,
it doesn't get old, it makes no sense to ask what its
physical size is and so on. So if it is an object, it's a peculiar kind of object.
It's an object that doesn't exist in space and time. It's not part of our spatial-
temporal world. Plato had a theory to account for this, which is
the famous theory of Forms. According to Plato, numbers are Forms, and
Forms are abstract, objectively existing objects. And the fact that they're not
spatio-temporal, they're not part of our world of space and time, didn't
bother Plato a bit. On the contrary, it confirmed him in his opinion that
reality is formal. The world of Forms is the reality of which our spatio-temporal
world is but a shadow, according to Plato. And according to Plato,
this accounts for the corrigibility of everything we get from our senses–
everything we see, everything we touch, everything we smell–the knowledge we get
from our senses is always open to revision. We might look again and
see something different. Whereas, formal knowledge is not. And
what that shows, according to Plato, is the superiority of our reason over our senses.
We can't see Forms, we can't touch Forms, but we can grasp them intellectually.
We can get knowledge of Forms, knowledge of arithmetic by
thinking. And according to Plato, that explains the characteristics of
mathematical truth: it's necessarily true, it's not contingently true. It's not
contingently true because it's not about the contingent world, the spatio-temporal
world. Now, the downside of this theory —the upside of this theory is that it
explains a lot about mathematical truth and mathematical knowledge. The
downside is that it requires us to believe in something that a lot of us
have trouble believing in, which is: an objectively existing world of Forms. And even if we could persuade
ourselves that such a world existed, we'd have an apparently insolvable problem
which is: How do we, so to speak, reach it? Given that we can't see it,
we have no sensory awareness of it, how do we bridge the apparently
unbridgeable gulf between that world and us? After all, we do exist in the spatio-
temporal world. So, for those reasons, a lot of philosophers,
I would say most philosophers, have had trouble persuading
themselves that Plato's world of Forms really exists. And therefore, that the
so-called mathematical realm that was supposed to be part of the world of
Forms, a lot of philosophers have trouble persuading themselves that that exists
as well. And these thoughts had occurred to Aristotle who was a pupil of Plato's,
who didn't believe in the world of Forms. He believed that mathematics is not
about objects in the mathematical realm, there is no such realm. Mathematics is about our world.
And okay, we can't see a number, but we might regard a number as
a property of things that we can see. So okay, numbers are not objects, but we
can understand them as features of objects. So we look at a field,
we see 4 cows, it's not that we see the cows and
then we see 4. It's that 4 is a property of that collection of cows that we see.
Well, philosophers since Aristotle have put forward powerful objections
to that way of looking at numbers, the most powerful of which were put by a
German mathematician-come-philosopher, Gottlob Frege, who was writing in
the 19th century. And he said look, when we know something about numbers,
we know it objectively. But numbers, Frege said, cannot objectively be
properties of other things. And the reason he said for that is that which
number belongs to a collection of things will depend upon how we conceptualize it.
So think of a deck of cards. A deck of cards has 52 cards in it. It has 4 kings,
it has 4 suits; 4 suits, 52 cards. Depending on whether—so we
have a deck of cards in front of us— depending on whether we're thinking
in terms of cards or of suits of cards, different numbers will belong to
that particular collection of things. So, does that collection of things have the
property 52? Or does it have the property 4? It has the property 52
if we're thinking of cards. It has the property 4
if we're thinking of suits. At a simpler level,
imagine a pair of shoes. It's 1 pair of shoes, but 2 shoes.
So, as an object, as a physical object, which number belongs to that? Is it the number
one or the number two? So, Frege says this is in general true that objects,
objectively, so to speak, do not have numbers as properties.
They acquire numbers as properties when we think of them in different ways.
And this is inconsistent with regarding mathematics as objective. So for those
reasons, the idea that numbers are properties of objects is one of those
ideas in philosophy that is, by a lot of philosophers, been regarded
as being refuted, refuted by Frege. We seem to be on the horns of a dilemma
where, when we solve certain problems about mathematics we encounter others.
If we do justice to the objectivity of mathematics, like Plato did, we seem lumbered with a kind of metaphysics,
a metaphysics of abstract objects. And when we asked ourselves
too deeply questions about what these abstract objects are supposed
to be and what the world of Forms is supposed to be, we find that we can't
give satisfactory answers to those questions. Where we can give satisfactory
answers to those questions we seem to be impaled on the other horn, which is,
we seem to have adopted a view which does away with the
objectivity of mathematics. Okay so, fast-forward now from the
Ancient world to the 18th century, Europe in the 18th century, and you come across
a great towering figure in philosophy, the German Immanuel Kant, whose great
work was the Critique of Pure Reason, published in the 1790s. Kant put forward
a theory of mathematics which became the most influential up until the 20th
century. And Kant did so in a way that grasped, as it were, one horn of this
dilemma, and put forward a theory that did away with the objectivity of
mathematics. Kant's thinking about mathematics starts with a question that
I didn't raise about mathematical knowledge but which is implicit in some of the
things I did say about it, which is this: If something is true mathematically, it's
necessarily true. And yet, mathematics works. 3 apples plus 2 apples really
is 5 apples. The world, as it were, seems to conform to the laws of
arithmetic. But the laws of arithmetic are not just true, they're necessarily true.
And Kant's question was: "How can we know something about the world which
is necessarily true?" So he introduced two distinctions which have since become
part of the technical vocabulary of philosophy. The first distinguishes two kinds of
sentence: an analytically true sentence and a synthetically true sentence. The difference is this: an analytically
true sentence is necessarily true. So, for example, "All bachelors are
unmarried". That's an analytic statement. It's an analytic statement because
it's true by definition. Compare it with the statement "All bachelors are unhappy".
It might be true that all bachelors are unhappy, but it's not necessarily true, it's not
part of the definition of a bachelor that bachelor is unhappy. But it is part of
the definition that a bachelor is unmarried. So the statement "All bachelors
are unmarried" is necessarily true because it's true by definition.
Kant called this an analytic statement as opposed to a synthetic statement
such as "All bachelors are unhappy". The reason he chose those particular terms
"analytic" and "synthetic" is to do with the question of whether you're dealing with
one concept or two. The idea here is that if you say "All bachelors are unhappy", you're making a synthesis of two quite
different concepts: the concept of being a bachelor & the concept of being unhappy.
If you say "All bachelors are unmarried", you're not synthesizing two
unrelated concepts, you're analyzing, so to speak, a feature of one concept. It's a
feature that — you know, what does the word 'bachelor' mean? It means unmarried man.
So if you analyze the concept 'bachelor', you can analyze it into the concepts
'unmarried' and 'male'. So that's why he said that's an analytic statement as
opposed to a synthetic statement. So there's another distinction he
drew which is in regard to how we know things to be true, and he used Latin
titles for this: 'a priori' and 'a posteriori'. Something is a priori known
if our knowledge of it is prior to any experience, any observation,
any testing. So again, we know that all bachelors are unmarried, we don't have
to do a survey, we don't have to do a test. We know that a priori, we know that prior
to any testing, any surveying, and whatever. Whereas, "All bachelors are
unhappy", if it's true at all, it's going to be true a posteriori, it's going to
be true on the basis of doing some empirical research. We know
that smoking causes cancer, we didn't always know that, but we do
know that now. Why do we know it? Because we've done experiments, we've done
tests, we've looked at people who smoke, we've made observations, and so on.
So "smoking causes cancer" is a posteriori. Now, think of those two
distinctions. It ought to be the case that the analytic goes with the a priori,
and the synthetic goes with the a posteriori. But what Kant said about mathematics
— and this is to do with it being necessarily true and true of the world —
is he says in mathematics we've got this curious hybrid. He said mathematics
is not analytically true, it's not true by definition that 5 + 7 = 12.
So that's a synthetic statement according to Kant. And yet it's a priori.
We don't have to do any experiments to find it out. So mathematics — and this,
according to Kant, is its great single feature — mathematics is synthetic a priori.
And his question was: How on earth how can we know things "synthetic a priori"?
And his answer was the whole system of metaphysics that he puts in the Critique
of Pure Reason, which is called "Transcendental Idealism", at the heart of
which is the idea that we don't know anything and cannot know anything
about things in themselves. We can only know–so he called those
things 'noumena'. We can only know things about what Kant called 'phenomena',
which are not things as they are in themselves but things as they appear to us. Things
as they appear to us, according to Kant, have been put through a kind of filter
which is the way we see the world. And mathematics, according to Kant,
is that filter. In other words, we don't get mathematics from the world,
we bring it to the world. So if you think of mathematics
as divided into two parts: geometry and arithmetic. Geometry, according to
Kant, is the spatial form through which we see the world. It's the spatial glasses,
as it were, that we look at the world. The world appears to us to be
three-dimensional Euclidean space. It's the world, the space described by the
system of geometry that we got all those thousands of years ago from the ancient
Greek geometer Euclid. And the reason those things are necessarily true,
they're true a priori according to Kant, is that we didn't get them from the
world, we brought them to the world. We look at the world
through those spectacles. Where does arithmetic come in? Where
do numbers come in? According to Kant, just as geometry is the form of our
spatial awareness, our spatial intuitions, arithmetic is the form of our temporal
intuitions. So at the heart of arithmetic is a sequence of numbers 1, 2, 3, 4, 5… It's a one-dimensional sequence, which
corresponds, according to Kant, to our experience of time. We experience space
as three-dimensional. We experience time as a one-dimensional sequence,
which is the sequence of numbers. So numbers are the way we organize time
one moment after another moment. Geometry is the way we organize space.
And put together, they give us the framework of the spatio-temporal world that we
experience. Everything we know is known about the spatio-temporal world. In other
words, everything we know, we know through those spectacles which we
ourselves have put to the world. Okay, well I've said that was enormously
influential theory of mathematics. It was opposed in the late 19th and early 20th
century by the man I mentioned earlier, Gottlob Frege & by a British philosopher
and mathematician, Bertrand Russell. What Frege and Russell together
sought to do was replace the Kantian view of mathematics with a view
that did justice to the objectivity of mathematics. And they went
right back to Plato, with this proviso: that they reverse the ancient Greek
priority about geometry and arithmetic. The ancient Greeks regarded geometry
as foundational, Russell and Frege regarded arithmetic as foundational.
And this is for two reasons. One is in the middle of the 19th century, alternatives to
Euclid's system of geometry were discovered: Riemann's system and Lobachevsky's
system. And what these systems did is they dropped the assumption
that parallel lines will never meet. In these systems of geometry, parallel
lines do meet. And what that means is that in these systems, space is curved. If you
think of the space of the outside of a globe, think of drawing two parallel lines,
they're going to meet at the top and meet at the bottom. So in these systems,
parallel lines can meet, which means that space is curved. And what really
threw the cat among the pigeons– it was bad enough for Kant's theory
that there were alternatives to Euclid because now the question arises: "well,
which pair of glasses should we wear?" And according to Kant's theory, we
shouldn't have a choice about that. But the cat was really put among the
pigeons with Einstein in his theory of relativity, according to which physical space
is Riemannian and not Euclidean. It's not just that you can — I mean,
Riemann in the middle of the 19th century invented the system of geometry,
as it were, just for the hell of it because he was a pure mathematician.
He wanted to see what would happen if you drop the parallel postulate. But according
to Einstein, it's not just of theoretical interest, the world is Riemannian, physical
space is curved. Now in the light of those developments, together with a
second development, which is that it was discovered in pure mathematics that you can
build geometry upon arithmetic and algebra. And so Frege and Russell regarded
arithmetic and not geometry as the foundational branch of mathematics. So for
them the central question was about number: "What is number?" And they weren't very
happy with Kant's answer to that, which is: Number is something inside our
heads that we bring to the world. They weren't very happy with that because it
makes arithmetic about what's inside our heads. Whereas, for Frege and Russell,
it was crucial that arithmetic is a body of objective knowledge. And so they
went back to Plato, into Platonism, it's objective knowledge about forms.
Now both of them, quite separately–and this is a remarkable fact–quite separately, both
of them had the same thought about that, which is: we can do justice to Plato's
formal theory of arithmetic — the idea that arithmetic is really about things
and these things are forms — we can do justice to that if we show that
mathematics and arithmetic in particular is just logic. So their view is called
"Logicism". And it's the view that arithmetic can be shown
to be a branch of logic. Now this, of course, only leads to
Platonism if you take a Platonic view of logical objects, which is exactly what
Frege and Russell did. According to Frege and Russell, logic is about forms,
and forms really exist. How do they make that plausible? They made that plausible
by bringing together two things that previously had not been brought
together, which were logic and arithmetic. It's been a hundred years now and
over that hundred years we've become accustomed to the idea that logic and
mathematics have got strong things in common with each other. But a
hundred years ago that wasn't the case. A hundred years ago, logic belonged to
the humanities. Logic was what you learned if you learned literature, poetry,
rhetoric. It was part of the humanities. You learned Aristotle's system
of logic. Learning logic went hand-in-hand with studying classics at Oxford.
Whereas, mathematics was what you learned if you were a scientist. And they were not
considered to have very much to do with each other. Logic was to do with language. It was
to do with using language to construct arguments, and the logic of Aristotle
tells you which of those arguments are valid arguments and which are not valid
arguments. Whereas mathematics gives you techniques that you can then use in
science. Frege and Russell brought them together with a particular theory of
number. And I think I've just got time to expound this theory. This theory makes use
of the notion of a "class", a class of objects. All right, and the way you get to that
notion is this: through language. So you start — I mean what they're doing is building a
bridge between Aristotle's theory of logic and the study of languages, on the
one hand, and arithmetic on the other. And the way that bridge works is this.
You start with propositions, with sentences, with what is analyzed in logic. Okay so take some some sentences: "Plato is wise", "Aristotle is wise", "Socrates is
wise". Those sentences all have the same form. Now you could capture that form
by replacing the name with a variable, with 'x'. So now you have "x is wise".
Now, a class is this. A class is all the things that would satisfy that sentence
if you replace the x with a name. If you do that, you've got the class of
wise people. So the class would have in it Aristotle, Plato, Socrates, all those
people who could replace the x in "x is wise". So the jargon that they came up with for
this was: "Plato is wise", "Socrates is wise", those are propositions, but "x is wise"
is a propositional function. And the class is the extension of the propositional
function. The class is all those wise things. Alright so now they used that
notion to talk about numbers. Numbers are classes according to Frege
and Russell. The number 4 is the class of all those things that have 4 members.
So there are 4 points on a compass: north, south, east and west. There are 4
Beatles: John, Paul, George and Ringo. Collect together all those things that
have 4 members, and that, according to Frege and Russell, is the number 4.
Now notice that this is very different to the old property theory. It's not that the
number 4 is a property of those collections. The number 4 is an object, but
it's a particular kind of object, it's a class. And they built a whole
system of logic on that notion of class. Then, and I'm not going to go into this
because you've probably had enough of all this sort of thing, but–
I will go into it if you want, but we'll leave that for the question period. In 1901, Russell discovered a problem
with that theory of classes which is called Russell's paradox. And he sent it to Frege. He said, look I've just
read your work, your work is very similar to mine it makes use of class theory, have you thought about this problem?
He showed the contradiction that arises in the theory of classes. Russell,
at that time, was quite confident that the problem could be overcome, poor old Frege had a nervous breakdown.
And after spending hospital, he came out and said he wasn't a Logicist anymore,
he didn't believe Logicism was true. Russell persevered with it, but came up
with a different view of logic which he got from his pupil Wittgenstein,
according to which logic is not the study of objectively existing forms. According to
Wittgenstein, there aren't any forms, this is a myth. There aren't any forms. What you've got is
language and ways of putting words together. And according to the rules for putting
words together, sometimes what you end up with is what's called a tautology. So, if I say that "It's raining outside", that's gonna be either true or false.
If I say "It's not raining outside", that's gonna be either true or false. But if
I say "Either it's raining or it's not raining", that can only be true. That is a
tautology. That is necessarily true. And so Russell took this notion from
Wittgenstein and said that's what mathematical propositions are, mathematical
propositions are just tautologies. The reason they're necessarily true
is exactly the same as the reason that "All bachelors are unmarried"
is necessarily true, they're true by definition. And so
Russell towards the end of his life said that if there was a god–notoriously
he didn't believe there was a god– but if there was a god, he said, the truths
of mathematics would have exactly the same profundity as the truth that
"A four-legged animal is an animal". And on that note, I'll finish.