Do you have a working definition of the term ‘truth’?
Well, in
mathematics there used to be one very precise definition. This was an idea
enunciated primarily by Hilbert which
stated that there ought to be a finite set of axioms in a formal mathematical logic
such that all mathematicians could agree on them as a starting point. And then,
in principle, it would be mechanically possible to grind away deducing all the consequences of these axioms using
the mechanical rules of mathematical logic. It was thought that this would, at
least in principle, get to all the truths of mathematics, though in practice of
course this would be a very slow process. But the idea stemmed from this very
formal notion of what a truth is, and from the belief that we could show that
mathematics gives absolute certainty – an absolutely black and white notion of
truth. So this is the idea of truth as being something that’s provable. This
was the Holy Grail, so to speak, of mathematics. And it corresponds to the
Platonic notion that mathematics is absolute truth – it comes from this idea of
a world of Platonic ideas, where mathematics is absolutely sharp, crystal
clear, and everything is black and white.
So the Holy Grail, for Hilbert and those like him, was
to find out what those axioms that underlay mathematics were, and to prove why
they were those axioms. Is that the case?
Yes.
Mathematicians believed that they had absolute truth. Hilbert believed that by translating
this into more precise terms, there
would be a small set of axioms that
all mathematicians could agree on, and then using the rules of formal logic,
this in principle would tell you everything that was true in mathematics. So,
the goal was, on the one hand, to find these axioms – which wasn’t believed to be too difficult –
and then, on the other hand, to convince people that these were all the axioms.
There was also another stage, which was attempting to prove to people that
these axioms would not lead to a contradiction.
Could you just give us, in layman’s terms, an example
of what such a set of axioms might be?
Well, the goal,
as I said, was to have axioms for all of mathematics, and a typical example of
that is Euclidean geometry which some of us learnt when we were
schoolchildren, and which is two thousand years old.
Now the
interesting thing about this project is that it imploded. It’s not that they
weren’t able to carry it out – it was that it was shown that in principle it was impossible to carry it
out. And this came from a very shocking piece of work, done initially by
Gödel in 1931, and I believe much
more convincingly, using different methods, by Turing in 1936, and in my own work I have tried to carry
on with this.
To see how this was done, you need to understand that there is another notion of truth in pure mathematics, according to which an argument is good if it would convince a colleague. And that, of course, is a much more traditional approach. Mathematicians do not, in practice, find formal proofs based on formal axiomatic systems of the kind that had been proposed by Hilbert. These kinds of proofs would actually be done best not by human beings, but by machines that would check if they were correct. These would be proofs with all the details filled in, that don’t use English or any other natural language. Instead they would use a very artificial language, like a computer programming language, in order to be completely precise. The logical steps in these proofs would be extremely minute. And the idea was that if you could actually carry this out and convince people that you had succeeded in dotting the ‘i’s and crossing the ‘t’s, this would substantiate the Platonic, philosophical claim that mathematics gives absolute truth. It would prove that, at least in the domain of beautiful ideas, there is an absolutely black and white criterion for truth, which is: provability based on a set of axioms that all mathematicians could agree on, using logical rules.
Now, most people
thought that this could be done. And
the big surprise was that it
couldn’t. The notion of truth, even
in pure mathematics, where it seems to be least elusive, in fact turns out to
be elusive. And that was quite a nasty surprise − people were very
shocked. Let me put it another way:
Mathematicians traditionally believe that the notion of truth in pure
maths is much clearer and more definite than in other domains of human activity
such as the law or even physics, the discipline to which, perhaps, it is
closest. Mathematics doesn’t deal with the real world,
it deals with the Platonic world of ideas, or the mind of God. It deals with very simple concepts like zero, one, two, three,
concepts that are much clearer, simpler and more straightforward than things in
the real world, which are rather elusive. So pure maths set out to make things
definite and to substantiate the claim that it provides absolute truth and that
truth equals provability. But the surprise
was that it can’t be done.
And that has sort of left the philosophy of mathematics hanging in midair since
the 1930s!
Could you just take us through how Gödel and Turing
showed that it couldn’t be done, and what you’ve taken from them and how you’ve
continued that line of thought?
Yes, I will do
that. But first I just want to say that the generation of mathematicians
and philosophers who were concerned with these ideas basically disappeared with
the Second World War. What the mathematical community in general did was that
they sort of brushed Gödel and Turing’s work under the carpet and just carried on as before. Human beings have
a great ability not to face disagreeable things and so most pure mathematicians
carried on as before and ignored all of this. It didn’t seem to have a
practical impact on their daily work. In principle, yes, pure maths was left
floating in the air. But in practice it looked like the mathematicians in each
individual field could sort of shrug
it off and say, ‘well in my field all of this does not apply, or not very much,
not in the questions that interest me in practice. So I’ll just carry on as before and hope for the best.’ And in
fact this has worked very beautifully.
Pure maths has progressed remarkably since 1931 and 1936 in spite of those very shocking results from a philosophical point
of view. So I’m unusual and my stance is quite controversial, because I believe,
though I cannot make a completely water-tight case for this, that this incompleteness that Gödel
found is really very serious. And I’ve carried on in the tradition of Gödel and in particular Turing, by trying to
build a case that this really is serious, and that it
means that you can’t just make a small change in the philosophy of mathematics. Because this little crack shatters the whole thing – it’s
like a crystal goblet where just one little scratch will make the whole
thing explode. So I’ve tried to build
the best case I can to show that this means that mathematics is completely
different from what people thought. And that the notion of absolute truth and Platonic idealism is not the whole story. I believe that
my results suggest that perhaps pure mathematics is actually quasi-empirical
(which is a term that Lakatos invented and which I
will come to later). And by this I don’t mean to say that pure mathematics is
an empirical science. I don’t say it’s the same as physics, which is the
closest discipline. But maybe it’s not as different as most people think. So that’s my view, and I’ve been trying to make
a case for this, and of course my ideas are very controversial, and I’m sort of
a minority of one. But I have to say even I, myself, am not entirely convinced;
that’s as far as I’m able to go in one lifetime of research!
So could you put all of this in context for us, so we
can understand Gödel’s incompleteness theory, how Turing’s work on computable
numbers developed that, and then how you have championed that lineage today?
Well, let’s start with Gödel. We will look at how Gödel shattered Hilbert’s dream and then how Turing did it. The way Gödel did it is by using a familiar paradox – the paradox of the liar, which comes from the Greek philosopher Epimenides. And the paradox of the liar can be very simply put by saying: ‘I’m lying’, or, ‘the statement I am making at this very moment is a lie’, or ‘this statement is false’ perhaps. ‘This statement is false’ might be the best way to put the paradox. And the problem with this is that it can be neither true nor false. Because it says it’s false, and so if it is false, then what it says corresponds to reality, so therefore it has to be true. But if it’s true, it has to be false, because it says it’s false. You see, you just flip back and forth because it can’t be true or false. Then Gödel does the following: He takes the paradox and instead of having the statement say, ‘this statement’s false’, he has it say, ‘this statement is unprovable.’ And here ‘unprovable’ means unprovable in that wonderful Theory of Everything for mathematics which all mathematicians could agree on, and which Hilbert wished to find. So even if Hilbert could come up with this theory, Gödel, as we will see, has already found a fatal flaw.
The subtle
change from a statement saying ‘I’m false’ to a statement saying ‘I’m
unprovable’ is really crucial. We go from a paradox to a profound meta-theorem on the power of reason. Here is how it works. There are two
possibilities with the phrase ‘I’m unprovable.’ It is
either provable or it’s unprovable. But if you can prove something that says
it’s unprovable, you’re proving something that’s false. And that would be terrible
because it would mean that your beautiful theory for all of mathematics in fact
proves false results. So most mathematicians eliminate this as a possibility
because they consider it to be just too awful − it would leave
mathematics in ruins. So let’s assume that it is not the case. The only alternative left is that we
cannot prove the statement: ‘I’m unprovable.’
Therefore what it says corresponds
to reality: it’s true but it’s unprovable. So you
have a hole, your theory is incomplete. You have a true assertion which is unprovable and so your system has not captured all the rules for deriving
mathematical truths. You do not have a
Theory of Everything for all of mathematics. Gödel’s basic idea is very simple, but the technical details are
extremely complicated. Using only the language of Peano
arithmetic, Gödel managed to make a very complicated assertion which actually
does indirectly succeed in saying that it’s unprovable. And that was the really
brilliant technical side to his work. So there’s a side of his proof which is
very deep and very simple, and
then there’s a very technical, very complicated side, which…
… Which will go right over our
heads!
Perhaps! That is the side that people find impossible to understand, because it’s so complicated and so technical. But the basic idea is very profound and it came from the fact that Gödel dared to think that Hilbert might be wrong, which nobody did, you see. Complicated or not, the fact remains that Gödel’s method shows that any system which Hilbert could propose would have a flaw because you can construct an assertion which says that it is unprovable according to those laws or from those axioms. And this shows that those axioms or laws can’t capture all of the fundamental principles and methods of reasoning in mathematics. And so mathematics cannot really implement Hilbert’s absolutely black and white, crystal clear notion of truth as formal proof. But unfortunately Gödel’s true, unprovable assertion looks bizarre. Nonetheless, people were terribly shocked by this – people like John von Neumann and Hermann Weyl wrote essays describing how Gödel had pulled the rug out from under mathematics. But they also said ‘well, this assertion is just a little bit bizarre, it is not the kind of thing you normally work with.’
Yet already in
1936 Alan Turing gave what I believe to be a much more significant proof of incompleteness. Of course he
was building on Gödel’s work −
the pioneer’s work is always the most difficult. How did
Turing prove that there are holes, that any
mathematical theory will leave out
things? He got that from a deeper phenomenon, which is uncomputability – that there are things which cannot be
calculated. There are lots of things in pure mathematics that can be defined, but
cannot be known well enough to calculate them. And if you can’t calculate them,
this means that you cannot prove them either. That’s because a
completely formal mathematical theory enables you in principle to mechanically deduce all the truths − by running through all the possible chains
of reasoning, starting from the fundamental principles, using the mechanical
rules of logic. So if you can always prove what the answer is, you can always
calculate what the answer is. Turing wrote a wonderful paper about all of this
called ‘On Computable Numbers.’ The numbers he’s talking about are not zero,
one, two, three, they’re not whole numbers. They’re numbers called ‘real
numbers,’ and that means they’re like lengths or distances. So these are
numbers like π, or e, or the
square root of two − numbers which you measure with infinite precision. Another
example of this might be what you need in order to locate a point on a line by
working out the distance from the point of origin. And what Turing asks is ‘are such numbers computable
or not? Can you calculate them digit by digit?’ If each number corresponds to a point in a line segment, say, then
what if you want to calculate the numerical value digit by digit? And Turing shows that there are points on the line which you cannot
calculate digit by digit. In fact, a great many real numbers are uncomputable.
Indeed, as I’ll now explain, you can show that the vast majority are uncomputable.
The key step is to give a definition of what a calculation is, and to be able to discriminate between real numbers which you can calculate and those that can’t be calculated. And that’s what Turing’s paper does. It turns out that some real numbers can, fortunately, be calculated digit by digit on a computer. Things go very well with numbers like π, or e, or the square root of two: you can calculate these as accurately as you wish – it is very easy to do so. But the strange thing − and this is sort of Turing’s version of Gödel’s incompleteness theorem (although he did not put it that way) − is that if you pick a real number at random, the chances are it will be uncomputable. And that begins to suggest that the problem of incompleteness is really serious. We are not just dealing with isolated cases. Almost all real numbers are uncomputable. In this domain the problem is everywhere. It’s not an isolated singularity as was the case with Gödel’s proof. So I believe that Turing’s work is a really fundamental step forward. (Although do bear in mind that I am giving you my version of Turing’s paper here and combining that with some ideas from Emile Borel.)
Turing also brings the notion of a machine into all of this – because he talks about computers and so on − and that makes everything sound very physical, as if there are physical limitations to what can be done. And that’s something that modern researchers have picked up: combining ideas from physics with ideas about computation. It’s a way of uniting pure maths with physics. And that’s quite a hot subject now; for example, it deals with things like quantum computing. So, Turing’s paper was very seminal, I think, but it’s taken a long time to see everything that was important in that paper and Turing himself did not fully appreciate all of the consequences of his own work.
Ok, so what have
I tried to do? Well, I’ve added another idea to the mix, which is the idea of
conceptual complexity. And this idea,
interestingly, can be traced back to
Leibniz in 1686. Conceptual complexity means the complexity of ideas. Leibniz
had a version of this, but the modern way of measuring conceptual complexity, is
to ask: ‘how big does a piece of computer software need to be in order to embody these ideas?’ Or, ‘how many bits of
software are needed?’ You can measure software in kilobytes, or megabytes, or
you can just measure it in bits. If the idea can be embodied in software, and
that in general is where my interests lie, then there is an algorithm, or
a computer programme, which can check or calculate that idea. For example, I
look at the size of the programme
needed to generate all the
theorems in a particular field – for instance, all the consequences of the
axioms using the mechanical rules of logic. Hilbert’s idea that this can be
done mechanically corresponds to the fact that you can actually write out
software to do it on a computer. And so I measure the complexity of a mathematical
theory by measuring the size, in bits, of the program that systematically
deduces all the consequences of the
fundamental principles. This is a way to measure the conceptual complexity of a theory, to measure the information content of your theory, to
measure, as it were, how many bits of axioms you have. We live in a world where everything
is digital: photos are digital, music is digital, video is digital. Nowadays,
everything is zeros and ones – everything is discrete. So, in principle, we can
use this as a way to measure the complexity of an idea. We are just looking for the amount of software needed to calculate
something, or to do something.
So the idea is to look at how many bits of information there are in a mathematical theory and
therefore how complicated it is. Set theory is a concrete example. We can
measure the complexity of set theory in terms of the amount of software needed
to calculate all the consequences of the axioms of set theory. Ok, so once you
do this, what I can show is that this measure has a certain
usefulness, because you can limit, in
some cases, the power of a theory by using this complexity measure I’ve just
explained. So I have a way, using Turing’s
ideas on computation, to measure complexity in terms of the size of an
algorithm – I talk about it in terms of ‘bits of complexity’ − zeros and
ones of complexity. And using this idea, I can limit the power of a mathematical theory by the number of bits of
information in the theory. So this, I believe, makes incompleteness seem even more natural than it does in Turing’s
work. Because I’m sort of saying:
‘if you want to prove more things, you have to put more information, or more
complexity, into your theory – even if that is your Theory of Everything for
mathematics.’
So is that what you describe as ‘irreducible
complexity’?
Well, not quite. I will get to that in a moment. First, I want to point out that we simultaneously have, on the one hand, our mathematical theory with only a finite amount of complexity, and on the other hand, the beautiful Platonic world of mathematical ideas, which has infinite complexity. So even if you believe in the Platonic world of ideas, which I guess I do, and even if you believe that truth is black and white, well, even then, only God, so to speak, has an infinite mind and can understand everything, we cannot. What we can know, and prove, down at our level, is limited. Even if you believe in the world of absolute truth where, you know, zero, one, two, three and so on exist in some sense and any property of these numbers will be either true or false, well, you must realise that down at our level what we can know is limited. Any mathematical theory has a finite amount of information, a finite amount of complexity, whereas the Platonic world of mathematical ideas, of pure maths, has infinite complexity, it has an infinite number of bits of information. And that’s why incompleteness is natural and inevitable.
And so what my
work suggests, at least to me, is that the way mathematics has to get around those
astonishingly pessimistic results from Gödel and Turing is by adding new
principles, by adding more complexity, by adding more information to the ground
rules, by adding more axioms to your theory.
I call this a ‘quasi-empirical’ view of mathematics, and it is related to a
movement now called experimental maths, in which, instead of using normal
proofs, you present numerical evidence for mathematical results, and this can
be quite strong, even if it doesn’t amount to a traditional proof. I believe my
work provides some theoretical justification
for doing maths like this. But this is very controversial, and I admit that I
don’t make a completely convincing case. I think my work points in this
direction, but much remains to be done.
So where does the concept of irreducible complexity
come in to all of this? How does that relate to this idea that you’ve been
explaining, about constantly creating new axioms to describe things?
Well, if you have to be constantly adding complexity to your mathematical theories, then you have to accept that the notion of proof is time-dependent, because our theories would be constantly changing. We would have to accept that pure maths is a little bit more like physics, like an empirical science. Because in physics, the principles you use are constantly changing. Whereas the normal notion of mathematics is that truth is static and eternal. Well, even if you believe that to be the case in the Platonic heaven where mathematical concepts reside, what we can know down here would not be static. We can only know a finite amount, and if we want that to grow, we have to add new principles, we have to make our theories more complicated. And this is a bit like what physicists do and it is what I am proposing. And I would argue, controversially, that you can find Gödel saying something along these lines.
Now, in relation to irreducible complexity, one of the key points in my argument was that any mathematical theory only has a finite number of bits of information, finite complexity, whereas the world of pure maths has infinite complexity, or an infinite number of bits of mathematical information. How do I show that the world of pure maths has infinite complexity? It’s easy to see that our mathematical theories have finite complexity, because otherwise human beings wouldn’t be able to work with them. But to show that the world of pure maths has infinite complexity is harder. And the nicest way I’ve found to do this involves something that I think Alan Turing would be delighted to know about, because my idea is based on his work. Turing is particularly famous for something called the Halting Problem. In fact, Turing is known for doing simultaneously two contradictory things. On the one hand, he creates, in a way, the notion of the computer as a mathematical concept. Now we call that a ‘Turing machine.’ This was before there was any computer technology, but it’s very clear that he had the idea of what a computer is. A computer is a very flexible, general-purpose digital calculating machine. You change the software, not the hardware, that’s why it’s so flexible. But simultaneously, Turing says that there are things that these machines can’t do. I talked about uncomputable numbers, but Turing also showed that there is something called the Halting Problem which cannot be settled. This is the question of whether a computer programme stops eventually, or not. And he shows there is no algorithmic or mechanical way to decide this automatically. This is a very fundamental, negative result. So Turing on the one hand creates the computing industry, in a way, and on the other hand he establishes a devastating limitation, from the point of view of pure maths, on what any computer can achieve. So, on the one hand, he giveth, on the other, he taketh away!
Anyway, the thing that I’ve come up with, which I’m proud of, and which I believe Turing would like, is a real number that I call the Halting Probability, which is like a pun on Turing’s Halting Problem. You write it like this: Ω. The Halting Problem is actually an infinite number of problems, because there are an infinite number of individual cases. The Halting Probability Ω puts it all in one package, by asking: ‘what is the probability that a computer programme that is picked at random will eventually come to a stop?’ We are talking about a self-contained computer programme which just starts grinding away and never asks for input (because if it did ask for input, whether it stopped or not would depend upon that input). This has to be a programme that just starts running and never asks for any more information, it just keeps going and then you ask whether it’ll go on forever or whether it’ll stop eventually. Now, of course, it may take a very, very, very long time. You see, this is pure maths, not a practical question. It’s a very fundamental philosophical question, but it’s not a practical question.
So let’s pick a
programme at random and ask what is the probability that it
will eventually stop? This combines the answer to all individual cases
of Turing’s Halting Problem in
one number, one numerical value, called Ω. The Halting Probability Ω would be zero if no programmes stopped, and it
would be one if every programme stopped. And some programmes finally come to a
halt, and others don’t, so the Halting Probability is actually greater than zero and less than one. Ω’s numerical value is actually rather elusive. Even
though mathematically this number is very easy to define – I just defined it in
words – even though it is
conceptually rather straightforward, it turns out that if you wanted to know
the numerical value of this number, if you actually wanted to calculate it,
this would be effectively impossible. Now remember, Turing’s original paper is
on computable numbers, he is interested in calculating real numbers – numbers
like distances that correspond to points on a line or like my Halting
Probability. And what is interesting about the Halting Probability is
that it is maximally uncalculatable. Turing already knew that there were uncomputable numbers – he gives an example in his paper.
But the Halting Probability is a
worst case. The way I like to explain this is: imagine that the numerical
value of the Halting Probability is written in binary, in base two, in zero/one bits. So there would be a binary point, a
decimal point, followed by a lot of digits, which would be zeros and ones. Every
one of those binary digits has got to be a zero or a one, in the same way that
in a normal number, if you write it in decimal, every digit has to be a zero,
one, two, three, four, five, six, seven, eight or nine. But with the Halting Probability,
the question of whether each bit is a zero or a one is very delicately
balanced. And whilst in the Platonic world of ideas it will precisely be either
a zero or a one, down here, on Earth, where we live, with our reasoning
ability, and our computing abilities, it
really just looks completely random.
The best way to describe it is to say that what we can know, using mathematical
methods available to human beings, using computational methods available to
human beings, is that each bit of this number is equally likely to be zero or
one. That’s all. It is like tossing a coin: it has got to be heads or tails,
but the result of each toss is a complete surprise. It has got to be one or the
other, but we can never tell in advance which. The funny thing is that in pure maths, all
truths should be necessary, and
whilst they might be in the Platonic world of ideas, down here, based on our
powers of reasoning, and our powers of computation, the numerical value of the
Halting Probability really just looks contingent.
Its bits really look random, or accidental. There is no structure or pattern
that we can see or appreciate, or that we can capture, with our methods of computation
or methods of proof. The bits of the
number Ω contain, in fact, an infinite amount of mathematical complexity,
because each bit of the numerical value of the Halting Probability is an independent, irreducible
mathematical fact. In other words, even if you knew the first billion bits, the next bit would still be a
big surprise. It would still look equally likely to be a zero or a one. Even if
you knew all the even bits, the odd bits would still be a big surprise. No
matter how much computing you do, you still can’t see any structure or discern any
pattern in the bits of this number. And essentially the only way of proving
what the numerical value of an individual bit is, is to add that as a new
axiom. That’s what I mean by irreducible
complexity. The bits of this number, Ω, are logically and computationally irreducible.
That means that essentially the only way to get those bits out of a
mathematical theory is to put them in directly as an axiom. But you can prove
anything by adding it as an axiom. And
that’s useless because you are not
using reasoning. So this is sort of a worst case for reasoning. This is a place where mathematical truth looks
accidental or contingent.
Assuming you are right about all this, then what are
the metaphysical, or philosophical, or epistemological implications for our
understanding of the world?
Well, that’s a
good question. This Ω number does exist. The real question is, ‘how typical is it?’ There
is the same problem with Gödel’s original proof, and also with Turing’s results.
And so we have these highly controversial questions: How much does all this
impact on the normal mathematics that normal mathematicians do? Where does it leave our traditional notion that
mathematical truth is absolutely black or white, and crystal clear? And I’m
assuming in this whole discussion that we believe in the Platonic world of
ideas, and that there truth is absolutely black and white, crystal clear and static and eternal. All that I
discuss in my work is what we can prove, what we can compute. So even if you
agree that that ideal world is out there, you can show, using mathematical
reasoning, that what we can know or prove is limited. Yet in traditional
philosophy, mathematics is supposed to be the example of where reason really
works. Most philosophers would love it if their discussions could have the same
degree of certitude that pure mathematics gives. Subjects are often seen as
being more ‘scientific’ to the extent that they become mathematical. So theoretical physics is considered to be an extremely ‘hard’
subject, and an extremely successful subject, because it’s very, very close
to pure maths. And other fields, like psychology for
example, which use maths very little
are viewed as ‘soft’ sciences. But maths, the hardest of the hard
sciences, proves that it is actually soft!
Ok, so where
does this leave the notion of truth, and how does this affect epistemology?
Well, I mentioned my tentative answer to this earlier. There is a beautiful
term coined by a Hungarian philosopher who immigrated to the
Another way to put it – and I believe that Gödel makes some remarks along these lines – is that there are things in physics, for example the Schrödinger equation, which are very important, and which no one would say were self-evident, a priori truths. Instead their justification is empirical and pragmatic. And I believe we should also allow this in pure maths. The normal notion of an axiom or a postulate in mathematics is that it has to be a self-evident truth. The pursuit of truth consists of breaking something complicated into smaller pieces until the pieces are so small, that they are self-evident and don’t need to be proved. The fundamental principles of maths are normally self-evident truths. They don’t need to be demonstrated. They’re simple enough that we can see that they are self-evident. If it were otherwise, a proof would never end. There would be an infinite regress if you kept trying to prove everything from something else.
So I guess I’m saying that even in pure maths, you should, perhaps, be prepared to add new, complicated principles, not because they are self-evident but because they are pragmatically justified and because they help you unify and understand a great many different mathematical phenomena. This means behaving much more like a physicist – that is why the term ‘quasi-empirical’ that Lakatos came up with is so good. But then pure maths would not give absolute certainty. What was accepted would be time-dependent, because you would be adding to the fundamental principles, and sometimes you might get it wrong and then have to backtrack.
So this is a
controversial view. I don’t see that there is such a tremendous difference between empirical science
and pure mathematics. Vladimir Arnold, a
Russian mathematician, put it this way: he said the only difference between
pure mathematics and physics is that the maths experiments are much cheaper! If you take this view it would remove
some of the certainty from mathematics, it would make maths a more human
endeavour and it would mean that sometimes mathematical theories might have to
be withdrawn if they were eventually proved wrong. Furthermore, mathematics
might break up into separate schools accepting contradictory axioms, and this
would make mathematics into a rather different kind of a game. Whilst we might
still believe in the Platonic world of ideas in principle, what we can know in
the human realm will always be limited by our human capabilities.
Finally, is there any kind of relationship between
this idea of an incompleteness theorem and, in physics, Heisenberg’s
uncertainty principle? They both seem to imply that there is a fundamental
impossibility at the heart of the intellectual pursuit. Is there a parallel
there?
That’s a very good question. A lot of us have been intrigued by that question. Heisenberg’s principle limits our capacity to measure the speed and position of a particle simultaneously, so if you measure one very precisely it makes the other one very uncertain. So I agree there is a parallel – Turing talks about there being a limit to what we can calculate, and I have tried to add to that the idea of informational limits. I do not know of a direct link between Gödel’s incompleteness theorem and Heisenberg’s uncertainty principle. But at a metaphorical, poetic, philosophical level there does seem to be a connection. The spirit of the time seems to be moving in the direction of limitations and randomness rather than determinism; toward unknowability and away from knowability.