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apologise)
/?'p?l.?.d?a?z/
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someone that you are
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. She apologized
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at 3.30 p.m. More examplesI
to apologize, but he just
at me.Trains may be
- we apologize for any
.Imagine the
of having to apologize.She apologized
for having to
at 3.30 p.m.He
he had been , and
apologized.
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A certain je ne sais quoi: French words and phrases used in English
by Liz Walter,
It is an odd irony that the more sophisticated your use of English is, the more likely you are to use French words and phrases. Or, to be more accurate, ones you know to be French – words such as ballet, au pair, abattoir, fiancé, café, and restaurant are so entrenched in
flower beard noun
a beard adorned with flowers And some of said beard-rockers are even turning it up a notch, painting trend on top of trend with what’s come to be known as ‘the flower beard.’58cv网址导航The philosophy of mathematics
Does mathematics need a philosophy?
Introduction
There is a philosophical doctrine known as bialetheism, with,
apparently, many adherents in Australia, who take the view that there
can be true contradictions. I do not wish to defend this view, but
nevertheless in the next 45 minutes I will be arguing, sincerely,
that mathematics both does and does not need a philosophy. Of course,
the apparent contradiction can in my case be resolved in a boringly
conventional way: the statement "Mathematics needs philosophy" has
at least two reasonable interpretations, and my contention is that
one of them is false and another is true. But before I tell you what
these interpretations are, I would like to say just a bit about what
I hope to achieve in this talk, since as the first speaker in the
series I have no precedents to draw on and have therefore had to
decide for myself what kind of talk would be appropriate.
First, as you can see, I am reading from a script, something
I would never do when giving a mathematics lecture. This, I fondly
imagine, is how philosophers do things, at least some of the time,
and it is how I prefer to operate when I need to choose my words
carefully, as I do today.
Secondly, I would like to stress that it is not my purpose to
say anything original. I have read enough philosophy to know that
it is as hard to be an original philosopher, at least if one wishes
to be sensible at the same time, as it is to be an original
mathematician. When I come across a philosophical problem, I
usually know pretty soon what my opinion is, and how I would begin
to defend it, but if I delve into the literature, I discover that
many people have had similar instincts and have worked out the defence
in detail, and that many others have been unconvinced, and that
whatever I think has already been labelled as some "-ism" or other.
Even if my tone of voice occasionally makes it sound as though I
think that I am the first to make some point, I don't. I have done
a bit of remedial reading and re-reading over the last couple of
weeks, but there are many large gaps and I will often not give credit
where it is due, and for that I apologize to the philosophers here.
Thirdly, I am very conscious that I am talking to an audience
with widely varying experience of mathematics and philosophy, so I
hope you will be patient if quite a lot of what I say about your
own discipline seems rather elementary and old hat.
Finally, I shall try to make the talk somewhat introductory.
Two weeks have not been enough to develop a fully worked-out
position on anything, so, rather than looking at one small issue
in the philosophy of mathematics, I shall discuss many of the big
questions in the subject, but none of them in much depth. I hope this
will give us plenty to talk about when the discussion starts - it
certainly ought to as many of these questions have been debated
for years.
If you ask a philosopher what the main problems are in the
philosophy of mathematics, then the following two are likely to come
up: what is the status of mathematical truth, and what is the nature
of mathematical objects? That is, what gives mathematical statements
their aura of infallibility, and what on earth are these statements
Let me very briefly describe three main
(overlapping) schools of thought that have developed in response to
these questions: Platonism, logicism and formalism.
The basic Platonist position is rather simple. Mathematical
concepts have an objective existence independent of us, and a
statement such as "2+2=4" is true because two plus two really does
equal four. In other words, for a Platonist mathematical statements
are pretty similar to statements such as "that cup is on the table"
even if mathematical objects are less tangible than physical ones.
Logicism is an attempt to justify our extreme confidence in
mathematical statements. It is the view that all of mathematics
can be deduced from a few simple and undeniably true axioms using
simple and undeniably valid logical steps. Usually these axioms
come from set theory, and they are supposed to form the secure
foundation on which the entire edifice of modern mathematics
rests. Notice that one can be a Platonist and a logicist at the
same time.
Formalism is more or less the antithesis of Platonism. One
can caricature it by saying that the formalist believes that mathematics
is nothing but a few rules for replacing one system of meaningless
symbols with another. If we start by writing down some axioms and
deduce from them a theorem, then what we have done is correctly
apply our replacement rules to the strings of symbols that
represent the axioms and ended up with a string of symbols that
represents the theorem. At the end of this process, what we know
is not that the theorem is "true" or that some actually existing
mathematical objects have a property of which we were previously
unaware, but merely that a certain statement can be obtained
from certain other statements by means of certain processes of
manipulation.
There is another important philosophical attitude to mathematics,
known as intuitionism, but since very few working mathematicians
are intuitionists, I shall not discuss it today. Let me just say
to those mathematicians who know a little about intuitionism that
certain aspects of it that seem very off-putting, such as the
rejection of the law of the excluded middle or the idea that a
mathematical statement can "become true" when a proof is found,
should not be dismissed as ridiculous: perhaps on a future
occasion we should have a debate here about whether classical
logic is the only logic worth considering. For what it is worth,
I myself am prepared to countenance other ones.
For the rest of this talk, I shall discuss some fairly specific
questions, in what I judge to be ascending order of complexity, with
the idea that they will give us a convenient focus for discussion
of more general issues of the kind I have been describing. But
before I do so, let me give you a very quick idea of where my own
philosophical sympathies lie.
I take the view, which I learnt recently goes under the name
naturalism , that a proper philosophical account of
mathematics should be grounded in the actual practice of mathematicians.
In fact, I should confess that I am a fan of the later Wittgenstein,
and I broadly agree with his statement that "the meaning of a word is
its use in the language". [Philosophical Investigations Part I section
43 - actually Wittgenstein qualifies it by saying that it is true
"for a large class of cases".] So my general approach to a philosophical
question in mathematics is to ask myself how a typical mathematician
would react to it, and why. I do not mean by this that whatever an
average mathematician thinks about the philosophy of mathematics
is automatically correct, but rather than try to make precise
what I do mean, let me illustrate it by my treatment of the
questions that follow.
1. What is 2+2?
The first question I would like to ask is this: could it make
sense to doubt whether 2+2=4? Let me do what I promised, and
imagine the reaction of a typical mathematician to somebody who
did. The conversation might run as follows.
Mathematician: Do you agree that 2 means the number after 1
and that 4 means the number after the number after 2?
Sceptic: Yes.
M: Do you agree that 2=1+1?
M: Then you are forced to admit that 2+2=2+(1+1).
M: Do you agree that addition is associative?
M: Then you are forced to admit that 2+(1+1)=(2+1)+1.
S: Yes I am.
M: But (2+1)+1 is the number after the number after 2, so it's 4.
S: OK I'm convinced.
The general idea of the above argument is that we have some
precise definitions of concepts such as 2, 4, "the number after"
and +, and one or two simple axioms concerning them, such as
the associativity of addition, and from those it is easy to
that 2+2=4. End of story.
That is, of course, roughly what I think, but before we move
on I would nevertheless like to try to imagine a world in which it
was natural to think that 2+2=5 and see what that tells us about our
belief that 2+2=4.
In such a world, physical objects might have less clear
boundaries than they do in ours, or vary more over time, and the
following might be an observed empirical fact: that if you put two
objects into a container, and then another two, and if you then
look inside the container, you will find not four objects but five.
A phenomenon like that, though strange, is certainly not a logical
impossibility, though one does feel the need for more details: for
example, if a being in this world holds up two fingers on one hand and
two on the other, how many fingers is it holding up? If you put
apples into a bag and then put no further apples into
the bag, do you have one apple? But then why not a tomato?
We are free to invent any answers we like to such questions if
we can somehow remain logically consistent, so here is a simple
suggestion. Perhaps in the strange world it is really the act of
enclosing objects in a container that causes what seems to us to
be peculiar consequences. It might be that this requires an
expenditure of energy so that the container doesn't just explode
the moment you put anything into it and there isn't the physical
equivalent of what economists call arbitrage. And yet, the apparent
duplication-machine properties of plastic bags and the like
might be sufficiently common for 2+2=5 to seem a more natural
statement than 2+2=4.
But what, one wants to ask, about our mental picture of numbers?
If we just
of two apples and then think of another
two, surely we are thinking of four apples, however you look at it.
But what should we say if we put that point to a being X from the
other world, and X reacted as follows?
X: I don't know what you're talking about. Look, I'm thinking
of two apples now. [Holds up three fingers from one hand.] Now I'm
thinking of two more. [Holds up three fingers from the other hand
making a row of six fingers.] The result - five apples.
Suppose that we were initially confused, but after a bit of
discussion came to realize that X was associating the apples not with the
fingers themselves but with the
gaps between them . After
all, between three fingers there are two gaps and between six fingers
there are five. At any rate this might seem a good explanation to us. But
perhaps X would be so used to a different way of thinking that it would
resist our interpretation. To X, holding up three fingers and saying "two"
could seem utterly natural: it might feel absolutely no need for a
one-to-one correspondence between fingers and apples.
Faced with such a situation, it is a tempting to take the
following line: what X is "really" doing is giving different names
to the positive integers. When X says "2+2=5", what this actually
means is "3+3=6", and more generally X's false sounding statement
that "a+b=c" corresponds to our true statement "(a+1)+(b+1)=c+1".
But should we say this? Or is it better to say that what X means
by addition is not our notion of addition but the more complicated
(or so we judge) binary operation f(m,n)=m+n+1? Or is it enough
simply to say that X uses a system of arithmetic that we can
understand and explain in terms of ours in more than one way?
These questions are bothersome for a Platonist, particularly one
who believes in direct reference, a philosophical doctrine I shan't
discuss here. If the word "five", as used by us, directly refers to
the number 5, then surely there ought to be a fact of the matter as to
whether the same word used by X directly refers to 5 or 6 or something
else. And yet there doesn't seem to be such a fact of the matter.
I think I will leave that question hanging since the world I
have just attempted to describe is rather fanciful and there are
many other ways of attacking Platonism.
2. The empty set.
My next question is whether there is such a thing as the empty
set. This question might seem more basic than the first, but if it
does then I put it to you that your mind has been warped by a century
of logicism, because there is, if you think about it, something
rather odd about the concept of a set with no elements. What, after
all, is a set? It is a collection of objects (whatever that means).
And to say that you have a collection of objects, except that there
are no objects, sounds like a contradiction in terms.
I have various questions like this, and normally I don't worry
about them. But they do cause me small problems when I lecture on
concepts such as sets, functions and the like. I will explain that
a set is a collection of objects, usually mathematical, but will
not go on to say what a collection is, or a mathematical object.
The empty set, I recently told the first-year undergraduates to
whom I am lecturing this term, is the set with no elements, but
I made no attempt to justify that there was such a set, and I'm
glad to report that I got away with it.
There are various ways that one could try to argue for the
existence of the empty set. For example, if it doesn't exist then
what is the intersection of the sets {1,2} and {3,4}? Or what is
the set of all natural numbers n such that n=n+1? These arguments
demonstrate that the empty set does indeed exist if one is
prepared to accept natural statements such as that the sets {1,2}
and {3,4} exist and that given any two sets A and B there is a
set C consisting of exactly the elements that A and B have in common.
So if you are going to doubt the existence of the empty set you
will probably find yourself doubting the existence of any sets
So let us consider the more general question: are there sets?
What exactly are we doing to the numbers 1 and 2 if we separate
them with a comma and enclose them in curly brackets? Similarly,
what is the difference between the number 1 and the set whose sole
element is 1?
Let us look at a simple problem that I recently set to the
first-years. I asked them whether there could be a sequence of
sets A1, A2, A3,... such that
for every n the intersection of the first n Ai is
non-empty but yet the intersection of all the Ai is
empty. The answer is yes and one example that shows it is to
let An={n,n+1,n+2,...}.
Why is this a suitable example? Well, the number n belongs to
all the first n Ai but if m is any number then m does
not belong to
Ai since it does not
belong, for example, to Am+1.
We could spell out this justification even more. Why, for
example, does n belong to the first n Ai? Well, n
belongs to Ai if and only if n&gti, so
n does indeed belong to all of A1,...,An.
Similarly, m does not belong to Am+1.
So an equivalent way to describe the above example is to say
that whatever (finite) number of conditions you impose on a number
n of the form n&gti, it will be possible for them to be
all to be satisfied, but there is no n that is greater than or
i. And the interesting thing about this
formulation is that it makes no explicit mention of sets. What's
more, it isn't just an artificial translation cooked up with the
sole purpose of not talking about sets. Rather, it reflects quite
accurately what actually goes on in our minds when we go about
proving what we want to prove about the sets.
So could it be that all that matters about the empty set is
something like this? Whenever you see the sentence "x is an
element of the empty set" it is false. More generally, could it
be that whenever you actually prove something about sets in a
normal mathematical context, one of the first things you do is
get rid of the sets. I had a good example of this recently when
proving in lectures that the equivalence classes of an equivalence
relation form a partition. If R is an equivalence relation on a set
A and x belongs to A, then the equivalence class of x is the set
E(x)={y in A: xRy}, but as the proof proceeded, every time I wrote
down a statement such as "z is an element of E(x)" I immediately
translated it into the equivalent and much simpler non-set-theoretic
statement xRz.
3. Subsets of the natural numbers
I think it would be possible to defend a position that set
theory could be dispensed with, at least when it involved sets
defined by properties. We could regard the expression
A={x:P(x)} not as actually denoting an object named A but as
being a convenient piece of shorthand. The statement "z belongs
to A", on this view, means nothing more than P(z). Similarly,
if B={x:Q(x)} then the statement "A cap B = emptyset"
means nothing more than that there is no x such that P(x)
But not all sets that crop up in mathematics, and I am
still talking about "ordinary" mathematics rather than logic
and set theory, are defined by properties. Often we talk about
sets in a much more general way, using sentences like, "Let A
be a set of natural numbers," and proving theorems such as
that there are uncountably many such sets. In such contexts
it is not as easy to dispense with the language of set theory.
And yet the sets we are supposedly discussing, general sets
of positive integers, are rather puzzling. My third question
an arbitrary set of positive
integers? Here I have in mind the sort of utterly general
set that cannot be defined, the infinite equivalent of a
subset of the first thousand integers chosen randomly. We
have a strong intuition that such sets exist, but why?
Let us look at the proof that there are uncountably many
sets of positive integers, and see what it tells us about our
attitude to sets in general. We start with an arbitrary sequence
A1,A2,A3,... of subsets of N,
and from those construct a new set A according to the rule
n is an element of A if and only if n is
not an element of An.
Then A is a subset of N not in the sequence. Since the
sequence we looked at was arbitrary,
sequence of
subsets of N exhausts all of them.
Why was A not in the sequence? Well, if it had been then
there would have had to be some n such that A=An.
But for each n we know that it belongs to A if and only if
belong to An, so A is not
the same set as An.
This argument shows a very basic property of the two
sets A and An - that they are not equal. And
yet even here I did not really reason about the sets themselves
and say some mathematical equivalent of, "Look, they're different."
Instead, I used the standard criterion for when two sets are equal:
A=B if and only if every element of A is an element
of B and every element of B is an element of A.
This tells me that in order to prove that A and An
are distinct I must find a positive integer m and show that either
m belongs to A but not to An
m belongs to An but not to A.
So, once again, what I seem to be focusing on is not so much
the sets themselves but statements such as "m is an element of A".
Does this mean that set-theoretic language is dispensable to
an ordinary mathematician? I think it often is, but I wouldn't
want to go too far - after all, I would certainly feel hampered
if I couldn't use it myself. Here is an analogy that I have not
had time to think about in any detail, so perhaps it won't stand
up, but let me float it anyway. A central project in philosophy
is to explain the notion of truth. What is it for a statement to
be true? There are some who hold that the word "true" adds very
little to our language: if we say that the sentence, "Snow is white"
is true, then what we have said is that snow is white, and that
is all there is to it. And yet the word "true" does seem to be
hard to avoid in some contexts. For example, it isn't easy to
think of a way to paraphrase the sentence, "Not all of what George
Bush says in the next week will be true" without invoking some
notion of a similar nature to that of truth. I think perhaps it
is similar for the language of sets - that it makes it much
easier to talk in generalities, but can be dispensed with
when we make more particular statements.
I must press on, but before I ask my next question, let me tell
you, or remind you, of three useful pieces of terminology.
4. Some terminology
The first is the phrase "ontological commitment", a phrase
associated with and much used by Quine. One of the standard tricks
that we do as mathematicians is "reduce" one concept to another -
showing, for example that complex numbers can be "constructed" as
ordered pairs of real numbers, or that positive integers can be "built
out of sets".
People sometimes use extravagant language to describe
such constructions, sounding as though what they are claiming is that
positive integers "really are" special kinds of sets.
Such a claim
is, of course, ridiculous, and probably almost nobody, when pressed,
would say that they actually believed it.
But another position, taken by many philosophers, is more appealing.
In describing the world, and in particular the rather problematic
abstract world of mathematics, it makes sense to try to keep one's
list of dubious beliefs to an absolute minimum. One example of a
belief that might be thought dubious, or at least problematic,
is that the number 5 actually exists. Questions about what exists
belong to the branch of philosophy known as ontology, a word
derived from the Greek for "to be"; and if what you say implies
that something exists, then you are making an ontological commitment.
One view, which I do not share, is that at least some ontological
commitment is implicit in mathematical language. But those who
subscribe to such a view will often seek to minimize their
commitment by reducing concepts to others. Such people may,
for example, be comforted by knowing that complex numbers can
be thought of as ordered pairs of real numbers, so that we
are not making any
worrying ontological
commitments when we introduce them than we had already made
when talking about the reals.
The second piece of terminology is the distinction between naive
and abstract set theory. A professional set theorist does not spend
time worrying about whether sets exist and what they are if they
do. Instead, he or she studies
of set theory, which
are mathematical structures containing things that we conventionally
call sets - rather as a vector space contains things that we
conventionally call vectors.
And just as, when we do abstract linear
algebra, we do not have to say what a vector is, or at least we do not
have to say any more than that it is an element of a vector space, so,
when we do set theory, we do not need to say what a set is, except to
say that it belongs to a model. (In an appropriate metalanguage one
could say that a model is a set, and sets, in the sense of the object
language, are its elements. But this is confusing and it would be more
usual to call the model a proper class.)
And to pursue the analogy
further, just as there are rules that tell you how to form new vectors
out of old ones - addition and scalar multiplication - so there are
rules that tell you how to form new sets out of old ones - unions,
intersections, power sets, replacement and so on. To get yourself
started you need to assume that there are at least some sets, so you
want an axiom asserting the existence of the empty set, or perhaps of
a set with infinitely many elements.
If you attempt to say what a set is, then you are probably
doing naive set theory. What I have just described, where sets
are not defined (philosophers would call the word "set" an
undefined primitive), is abstract set theory. Notice that
as soon as you do abstract set theory, you do not find yourself
thinking about the actual existence or nature of sets, though
you might, if you were that way inclined, transfer your worries
into the meta-world and wonder about the existence of the model.
Even so, when you were actually doing set theory, your activity
would more naturally fit the formalist picture than the Platonist
The distinction between naive and abstract set theory gives
one possible answer to the question "What is an arbitrary set
of natural numbers?" The answer is, "Don't ask." Instead, learn
a few rules that allow you to build new sets out of old ones
(including unions, intersections and the diagonal process we
have just seen) and make it all feel real by thinking from
time to time about sets you can actually define such as the
set of all primes - even if in the end the definition is
more important than the set.
Another distinction, which I introduce because it may
well have occurred to those here who have not previously
come across it, is one made by Rudolf Carnap between what
he called internal and external questions. Suppose I ask
you whether you accept that there are infinitely many
primes. I hope that you will say that you do. But if I
then say, "Ah, but prime numbers are positive integers
and positive integers are numbers and numbers are mathematical
objets so you've admitted that there are infinitely many
mathematical objects," you may well
feel cheated. If you do, the chances are that you will
want to say, with Carnap, that there are two senses of
the phrase "there exists". One is the sense in which it
is used in ordinary mathematical discourse - if I say that
there are infinitely many primes I merely mean that the
normal rules for proving mathematical statements license
me to use appropriate quantifiers. The other is the more
philosophical sense, the idea that those infinitely many
primes "actually exist out there". These are the
uses respectively.
And it seems, though not all philosophers would agree,
that this is a clear distinction, and that the answers
you give in the internal sense do not commit you to
any particular external and philosophical position.
In fact, to many mathematicians, including me, it is not
altogether clear what is even meant by the phrase "there
exists" in the external sense.
5. Ordered pairs.
I will not spend long over my next question, since I have
discussed similar issues already, but it is one of the simplest
examples of the slight difficulty I have when lecturing about
basic mathematical concepts from the naive point of view. The
question is, what is an ordered pair?
This is what I take to be the standard account that a
mathematician would give. Let x and y be two mathematical
objects. Then from a formal point of view the ordered pair
(x,y) is defined to be the set {{x},{x,y}}, and it can be
checked easily that
{{x},{x,y}}={{z},{z,w}} if and only if
x=z and y=w.
Less formally, the ordered pair (x,y) is a bit like the
set {x,y} except that "the order matters" and x is allowed to
Contrast this account with the way ordered pairs are
sneaked in at a school level. There, the phrase "ordered
pair" is not even used. Instead, schoolchildren are told that
points in the plane can be represented by coordinates, and
that the point (x,y) means the point x to the right and y up
from the origin. It is then geometrically obvious that
(x,y)=(z,w) if and only if x=z and y=w.
Pupils who are thoroughly used to this idea
will usually have no difficulty accepting later on that they
can form "coordinates" not just out of real numbers but also
out of elements of more general sets. And because of their
experience with plane geometry, they will take for granted
that (x,y)=(z,w) if and only if x=z and y=w, whether or not
you bother to spell this out as an axiom for ordered pairs.
In other words, it is possible to convey the idea of an
ordered pair in a way that is clearly inadequate from the
formal point of view but that does not seem to lead to any
problems. One can quite easily imagine an eminent physicist
successfully using the language of ordered pairs without
knowing how to formalize it.
It is clear that what matters in practice about ordered
the condition for when two of them
are equal. So why does anybody bother to "define" the ordered
pair (x,y) as {{x},{x,y}}? The standard answer is that if you
want to adopt a statement such as
(x,y)=(z,w) if and only if x=y and z=w
as an axiom, then you are obliged to show that your axiom
is consistent. And this you do by constructing a model that
satisfies the axiom. For ordered pairs, the strange-looking
definition (x,y)={{x},{x,y}} is exactly such a model. What
this shows is that ordered pairs can be defined in terms of
sets and the axiom for ordered pairs can then be
deduced from the axioms of set theory. So we are not making
new ontological commitments by introducing ordered pairs, or
being asked to accept any new and unproved mathematical beliefs.
This account still leaves me wanting to ask the following
question. Granted, the theory of ordered pairs can be reduced
to set theory, but that is not quite the same as saying that
an ordered pair is "really" a funny kind of set. ( That
view is obviously wrong, since there are many different
set-theoretic constructions that do the job equally well.)
And if an ordered pair isn't really a set, then what is it?
Is there any way of doing justice to our pre-theoretic notion
of an ordered pair other than producing this rather artificial
translation of it into set theory?
I don't think there is, at least if you want to start
your explanation with the words, "An ordered pair is". At
least, I have never found a completely satisfactory way of
defining them in lectures. To my mind this presents a pretty
serious difficulty for Platonism. And yet, as I have said, it
doesn't really seem to matter to mathematics. Why not?
I would contend that it doesn't matter because it
matters what a mathematical object is,
or whether it exists. What
matter is the
set of rules governing how you talk about it - or perhaps
I should say, since that sounds as though "it" refers to
something, what matters about a piece of
mathematical
terminology
is the set of rules governing its use.
In the case of ordered pairs, there is only one rule that
matters - the one I have mentioned several times that tells
us when two of them are equal (or, to rephrase again, the
one that tells us when we are allowed to write down that
they are equal, substitute one for another and so on).
I said earlier that I like to think about actual
practice when I consider philosophical questions about
mathematics. Another useful technique is to think what
you would have to program into a computer if you wanted
it to handle a mathematical concept correctly. If the concept
was an ordered pair, then it would be ridiculous
to tell your computer to convert the ordered pair
(x,y) into the set {{x},{x,y}} every time it came across it.
Far more sensible, for almost all mathematical contexts, would
be to tell it the axiom for equality of ordered pairs. And if
it used that axiom without a fuss, we would be inclined to
judge that it understood the concept of ordered pairs, at
least if we had a reasonably non-metaphysical idea of
understanding - something like Wittgenstein's, for example.
This point applies much more generally. I have sometimes
read that computers cannot do the mathematics we can because
they are finite machines, whereas we have a mysterious access
to the infinite. Here, for example, is a quotation from the famous
mathematician Alain Connes:
... this direct access to the infinite which characterizes
Euclid's reasoning [in his proof that there are infinitely many primes],
or in a more mature form G&del's, is actually a trait of the living
being that contradicts the reductionist's model.
But, just as it is not necessary to tell a computer what an
ordered pair is, so we don't have to embed into it some "model
of infinity". All we have to do is teach it some syntactic rules
for handling, with care, the
infinity - which is
also what we have to do when teaching undergraduates. And, just as
we often try to get rid of set-theoretic language when talking about
sets, so we avoid talking about infinity when justifying statements
that are ostensibly about the infinite. For example, what
Euclid's proof actually gives is a recipe for extending any finite
list of primes. To take a simpler example, if I prove that there are
infinitely many even numbers by saying, "2 is even and if n is even
then so is n+2", have I somehow exhibited infinite mental powers? I
think not: it would be easy to programme a computer to come up with
such an argument.
6. Truth and provability.
I can date my own conversion from an unthinking childhood
Platonism from the moment when I learnt that the continuum hypothesis
was independent of the other axioms of set theory. If as apparently
concrete a statement as that can neither be proved nor disproved, then
what grounds can there be for saying that it is true or that it is
But if you think there is no fact of the matter either way
with the continuum hypothesis, then why stop there? What about the
axiom of infinity - that there is an infinite set? It doesn't follow
from the other axioms of set theory, and nor, it seems, does its
negation. So why should we believe it?
Surely not because of some
view that the universe is infinite in extent, or infinitely
divisible. What would that show anyway? There would still be the
problem of applying those funny curly brackets. As I have said, even the
axiom that the empty set exists is hard to justify if one interprets
it realistically.
So I am driven to the view that there isn't much to mathematical
truth over and above our accepted procedures of justification - that
is, formal proofs. But something in me still rebels against
the intuitionists' idea that a statement could
true when a proof is found, and I'm sure most mathematicians agree
with me. So my next question is why, and I would like to look
at a few concrete examples.
Here is one that intuitionists like: consider the
statement "somewhere in the decimal expansion of pi there is
a string of a million sevens". Surely, one feels, there is
a fact of the matter as to whether that is true or false,
even if it may never be known which.
What is it that makes me want to say that the long string of
sevens is definitely either there or not there - other than a general
and question-begging belief in the law of the excluded middle? Well,
actually I am tempted to go further and say that I believe that the
long string of sevens
there, and I have a definite
reason for that stronger belief, which is the following. All the
evidence is that there is nothing very systematic about the sequence
of digits of pi.
Indeed, they seem to behave much as they would if
you just chose a sequence of random numbers between 0 to 9. This hunch
sounds vague, but it can be made precise as follows: there are various
tests that statisticians perfom on sequences to see whether they are
likely to have been generated randomly, and it looks very much as though
the sequence of digits of pi would pass these tests. Certainly the
first few million do.
One obvious test is to see whether any given
short sequence of digits, such as 137, occurs with about the right
frequency in the long term. In the case of the string 137 one would
expect it to crop up about one thousandth of the time in the decimal
expansion of pi.
If after examining several million digits we found
that it had in fact occurred a hundredth of the time, or not at all,
then we would be surprised and wonder whether there was an
explanation.
But experience strongly suggests that short sequences in the
decimal expansions of the irrational numbers that crop up in
nature, such as pi, e or the square root of 2,
with the correct frequencies. And if that is so, then we would
expect a million sevens to occur in the decimal expansion
of pi about 10-1000000 of the time - and it is
of course no surprise that we will not actually be able to
check that directly. And yet, the argument that it does
eventually occur, while not a proof, is pretty convincing.
This raises an interesting philosophical question.
A number for
which the short sequences of digits occur with the right frequencies
normal . Artificial examples have been constructed
of normal numbers, but there is no naturally occurring number that is
known to be normal. Perhaps the normality of pi is not just an
unsolved problem but actually an unprovable theorem. If so, then it is
highly unlikely that we shall ever find an abstract argument that
shows that the expansion of pi contains a million sevens in a row, and
direct calculation of the number of digits that would be necessary to
verify it "empirically" is out of the question. So what, then, is the
status of the reasonable-sounding heuristic argument that pi contains
a million sevens in a row, an argument that convinces me and many
This question raises difficulties for those who are too
ready to identify truth and provability. If you look at actual
mathematical practice, and in particular at how mathematical
beliefs are formed, you find that mathematicians have opinions
long before they have formal proofs. When I say that I think
pi almost certainly has a million sevens somewhere in its
decimal expansion, I am not saying that I think there is
almost certainly a (feasibly short)
assertion - perhaps there is and perhaps there isn't. So it
begins to look as though I am committed to some sort of
Platonism - that there is a fact of the matter one way or
the other and that that is why it makes sense to speculate about
There is an obvious way to try to wriggle out of this difficulty,
but I'm not sure how satisfactory it is. One could admit that a simple
identification of truth with provability does not do justice to
mathematical practice, but still argue that what really matters about
a mathematical statement is not some metaphysical notion of truth, but
rather the conditions that have to hold to make us inclined to assert
it. By far the most important such condition is the existence of a
formal proof, but it is not the only one. And if I say something like,
"pi is probably normal", that is just a shorthand for, "there is a
convincing heuristic argument, the conclusion of which is that pi is
Of course, a move like this leaves very much open the
question of which informal arguments we find convincing and why. I
think that is an important philosophical project, but not one I
have carried out, or one that I would have time to tell you about
now even if I had.
Actually, it is closely related to another interesting question, a
mathematical version of the well-known philosophical problem of
induction. A large part of mathematical research consists in spotting
patterns, making conjectures, guessing general statements after
examining a few specific instances, and so on. In other words,
mathematicians practise induction in the scientific as well as
mathematical sense. Suppose, for example, that f is a complicated
function of the positive integers arising from some research problem
and that the first ten values it takes are 2, 6, 14, 24, 28, 40, 42,
66, 70, 80. In the absence of any other knowledge about f, it is
reasonable to guess that it always takes even values, or that it is an
increasing function, but it would be silly to imagine that f(n) is
always less than 1000. Why? I think the beginning of an answer is that
any guess about f should be backed up by some sort of heuristic
argument. In this case, if we have in the back of our minds some
picture of a "typical function that occurs in nature" then we might be
inclined to say that the likelihood of its first ten values being even
or strictly increasing just by chance is small, whereas the likelihood
of their all being less than 1000 is quite high.
Let me return to the question of why it seems so obvious
that there is a fact of the matter as to whether the decimal
expansion of pi contains a string of a million sevens. In the
back of many minds is probably an argument like this. Since
pi almost certainly
normal, if we look instead
for shorter strings, such as 137, then we don't have to look
very far before we find them. And in principle we could do the
same for much longer strings - even if in practice we certainly
can't. So the difference between the two situations is
not mathematically interesting and should not have any
philosophical significance.
Now let me ask a rather vague question: what is interesting
about mathematical theorems that begin "for every natural number
n"? There seem to me to be two attitudes one can take. One is
that a typical number of the order of magnitude of, say,
1010100, will be too large for us to
specify and therefore isn't really anything more to us than a
purely abstract n. So the instances of a theorem that starts
"for all n" are, after a certain point, no more concrete than
the general statement, the evidence for which consists of a
certain manipulation of symbols, as a formalist would contend.
So, in a sense the "real meaning" of the general theorem is
that it tells us, in a succinct way, that the small "observable"
instances of the theorem are true, the ones that we might wish
to use in applications.
This attitude is not at all the one taken by most pure mathematicians,
as is clear from a consideration of two unsolved problems. Goldbach's
conjecture, that every even number over 4 is the sum of two primes,
has been verified up to a very large number, but is still regarded
as completely open. Conversely, Vinogradov's three-primes theorem,
that every sufficiently large odd integer is the sum of three
primes, is thought of as "basically solving" that problem,
even though in the current state of knowledge "sufficiently large"
means "at least 1013000" which makes checking the
remaining cases way beyond what a computer could do. This last
example is particularly interesting since to date only 79 primes
are known above 1013000 (or even a third of 1013000)
are known. So the theorem has almost no observable consequences.
In other words, there are two conclusions you can draw from
the fact that very large integers are inaccessible to us. One
is that what actually matters is small numbers, and the other
is that what actually matters is abstract statements.
One small extra comment before I move to a completely new
question. Another conjecture that seems almost certainly true is the
twin-primes conjecture - that there are infinitely many primes p for
which p+2 is also prime. This time the heuristic argument that backs
up the statement is based on the idea that the primes appear to be
"distributed randomly", and that a sensible-looking probabilistic
model for the primes not only suggests that the twin-primes conjecture
is true but also agrees with our observations about how often they
occur. But I find that my feeling that there must be a fact of the
matter one way or the other is less strong than it was for the sevens in
the expansion of pi, because no amount of finite checking could, even
in principle, settle the question. The difference is that the pi
statement began with just an existential quantifier, whereas "there
are infinitely many" gives us "for all" and then "there exists". On
the other hand, there does seem to be a fact of the matter about
whether there are at least n twin primes, for any n you might choose
to specify.
But now I am talking about very subjective feelings, so
it is time to turn to my last question.
7. The axiom of choice
I mentioned earlier that the status of the continuum hypothesis
convinced me that Platonist views of mathematical ontology and truth
could not be correct. Instead of discussing that, let me ask a similar
question. Is the axiom of choice true?
By now it probably goes without saying that I don't think that
either it or its negation is true in any absolute, transcendent,
metaphysical sense, but many philosophers disagree. I recently
read an article by Hilary Putnam in which he ridiculed the idea
that one could draw any philosophical conclusions from the
independence of the continuum hypothesis. But, as nearly always
happens in philosophy, I emerged with my beliefs intact, and
will now try to do what he thinks I can't with the axiom of choice.
Consider the following statement, which is an infinitary analogue
of a famous theorem of Ramsey.
Let A be a collection of infinite subsets of the natural numbers.
Then there is an infinite set Z of natural numbers such that either
infinite subsets belong to A or none of them do.
As it stands, that statement is false (or so one usually says)
because it is quite easy to use the axiom of choice to build a
counterexample. But there are many mathematical contexts in which
the result could be applied if only it were true, and actually for
those contexts - that is, for the specific instances of A that crop up
"in nature", as mathematicians like to say - the result
true. In fact, there is a precise theorem along these lines,
which comes close to saying that the only counterexamples are
ridiculous ones cooked up using the axiom of choice. So in a
way, the statement is "basically true", or at least true whenever
you care about it. In this context the axiom of choice is a minor
irritant that forces you to qualify your statement by putting some not
very restrictive conditions on A.
There are many results like this. For example, not every
function is measurable but all the ones that you might actually
want to integrate are, and so on.
Now let's consider another statement, the infinite-dimensional
analogue of a simple result of finite-dimensional linear
Let V be an infinite-dimensional vector space over R and let
v be a non-zero vector in V. Then there is a linear map f from
V to R such that f(v) is non-zero.
To prove this in a finite-dimensional context you take the
vector v, call it v1, and extend it to a basis
v1,...,vn. Then you let f(v1)=1
and f(vi)=0 for all other i.
For an infinite-dimensional space, the proof is exactly
analogous, but now when you extend v1 to a basis,
you have to continue transfinitely, and since each time you
are choosing a vi you are not saying how you did
it, you have to appeal to the axiom of choice. And yet it
seems unreasonable to say that you can't make the choices
just because you can't specify them - after all, you can't
specify the choices you make in the finite-dimensional
context either, and for the same reason, that nothing has
been told to you about the vector space V.
If you are given an explicit example of V then the picture
changes, but there is still a close analogy between the
two situations. Sometimes there is a fairly obvious choice
of the function f, but sometimes there is no canonical
way to extend v to a basis. Then, what rescues you if
V is finite-dimensional may be merely that you have to
make 10 trillion ugly non-canonical choices rather than
infinitely many.
But by and large, for the vector spaces that matter, it
is clear that a function f can be found, so this time it
would be the
of the axiom of choice that
is a minor irritant, telling you that you have to apologize
for your theorem by confessing that it depends on the axiom.
So for good reasons - and this is what I would like to
stress - we sometimes dismiss the consequences of the axiom
of choice and we sometimes insist on them. And in both cases
our governing principle is nothing to do with anything like
truth, but more a matter of convenience. It is as though
when we talk about the world of the infinite, we think of
it as a sort of idealization of the finite world we actually
inhabit. If the axiom of choice helps to make the infinite
world better reflect the finite one then we are happy to
use it. If it doesn't then we describe its consequences
as "bizarre" and not really part of "mainstream mathematics".
And that is why I do not believe that it is "really" true
or "really" false.
8. Concluding remarks.
It may seem as though I have ignored the title of my
talk, so here is what I mean when I say that mathematics
both needs and does not need a philosophy.
Suppose a paper were published tomorrow that gave a
new and very compelling argument for some position in the
philosophy of mathematics, and that, most unusually, the
argument caused many philosophers to abandon their old beliefs
and embrace a whole new -ism. What would be the effect on mathematics?
I contend that there would be almost none, that the development
would go virtually unnoticed. And basically, the reason is
that the questions considered fundamental by philosophers
are the strange, external ones that seem to make no difference
to the real, internal business of doing mathematics. I can't
resist quoting Wittgenstein here:
A wheel that can be turned though nothing
else moves with it, is not part of the mechanism.
Now this is not a wholly fair comment about philosophers
of mathematics, since much of what they do is of a technical
nature - attempting to reduce one sort of discourse to another,
investigating complicated logical systems and so on. This may
not be of much relevance to mathematicians, but neither are
some branches of mathematics relevant to other ones. That
does not make them unrespectable.
But the point remains that if A is a mathematician who
believes that mathematical objects exist in a Platonic sense, his
outward behaviour will be no different from that of his colleague B
who believes that they are fictitious entities, and hers in turn
will be just like that of C who believes that the very question
of whether they exist is meaningless.
So why should a mathematician bother to think about philosophy?
Here I would like to advance a rather cheeky thesis: that modern
mathematicians are formalists, even if they profess otherwise,
and that it is good that they are.
This is the sort of evidence I have in mind. When mathematicians
discuss unsolved problems, what they are doing is not so much
trying to uncover the truth as trying to find proofs.
Suppose somebody suggests an approach to an unsolved problem
that involves proving an intermediate lemma. It is common to
hear assessments such as, "Well, your lemma certainly looks
true, but it is very similar to the following unsolved problem
that is known to be hard," or, "What makes you think that the
lemma isn't more or less equivalent to the whole problem?"
The probable truth and apparent relevance of the lemma are
basic minimal requirements, but what matters more is whether
it forms part of a realistic-looking research strategy, and
what that means is that one should be able to imagine, however
that involves it.
I think that most successful mathematicians are very much
aware of this principle, even if they don't bother to articulate
it. But I also think that it is a good idea to articulate it -
if you are doing research, you might as well have as clear and
explicit an idea as possible of what you are doing rather than
groping about and waiting for that magic inspiration to strike.
And it is a principle that sits more naturally with formalism
than with Platonism.
I also believe that the formalist way of looking at
mathematics has beneficial pedagogical consequences. If you are
too much of a Platonist or logicist, you may well be tempted
by the idea that an ordered pair is
funny kind of set - the idea I criticized earlier. And if
you teach that to undergraduates, you will confuse them
unnecessarily. The same goes for many artificial definitions.
What matters about them is the basic properties enjoyed by the
objects being defined, and learning to use these fluently and
easily means learning appropriate replacement rules rather than
grasping the essence of the concept. If you take this attitude
to the kind of basic undergraduate mathematics I am teaching
this term, you find that many proofs write themselves - an
assertion I could back up with several examples.
So philosophical, or at least quasi-philosophical,
considerations do have an effect on the practice of mathematics,
and therein lies their importance. I have mentioned some
other questions I find interesting, such as the problem
of non-mathematical induction in mathematics, and I would
justify those the same way. And that is the sense in which
mathematics needs philosophy.
Brief additions in response to the discussion after the talk
1. Thomas Forster informed me that Russell and Whitehead
took roughly the view of ordered pairs that I advocate - treat
them as an undefined primitive with a simple rule for equality.
So perhaps I did logicists an injustice (unless they felt that
the construction of ordered pairs out of sets was significant
progress).
2. It was pointed out by Peter Smith that I had blurred the distinction
between a pure formalism - mere pushing around of symbols - and
"if-then-ism", the view that what mathematicians do is explore the
consequences of axioms to obtain conditional statements (if this set
of axioms is true, then this follows, while this other set implies
such and such else), but nevertheless statements with a definite
content over and above the formalism. I don't know exactly where I
stand, but probably a bit further towards the purely formal end than
most mathematicians. See my page on
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