Crashing the Barriers
Does it really matter if there are some things that science will never
solve? Ian Stewart thinks not
"We must know. We shall know." So said
David Hilbert, one of the leading
mathematicians at the turn of the century. Hilbert was gung ho about the
future of mathematics. No-go areas should not exist, he believed, and he
even had the outlines of a program to prove it. Yet within a few years, Hilbert's
dream lay in ruins -a young logician named
Kurt Gödel had proved that
some mathematical questions simply don't have answers.
What Gödel showed in 1930 was that any logical system
rich enough to model mathematics will always have
problems. For instance , it is impossible to prove that mathematics contains
no logical inconsistencies. Of course, you can deal with any particular insoluble
problem by adding a new mathematical rule, but a new insoluble problem will
always appear in the patched-up system.
Demise of science
At first sight the connection between Gödel's arcane
mathematical examples and the real world is not self-evident. But although
experimental science is about reality, theoretical science is about ideas,
and most of those ideas depend crucially on mathematical proof. So limits
to mathematics might well translate into limits to scientific theories.
Take the example of a toy train : you can't predict the path
of a toy train on some model railway layouts because of another famous insoluble
problem in mathematics, Alan
Turing's Halting Problem, which he described several decades ago in the
context of computational theory.
If you're using a computer for a task such as wordprocessing,
you would expect the machine to do what it was asked and then stop, ready
for the next task. However, programs can get "hung up" in infinite loops,
doing the same thing over and over again.
Turing, One of the fathers of modern computing, asked whether
it was possible to predict whether a program would eventually terminate or
go on for ever. He devised a model for the computing process which he called
a Turing machine consisting of a central
processing unit to do all the calculations, a program, and as much memory
as you need. Turing proved that within such a framework no mathematical theory
can predict in advance whether a given computation will ever stop.
He did this by assuming that a program that could do the job
existed, and then proving that this would lead to a logical inconsistency.
His argument was roughly as follows. Call the imaginary predictor program
A. Set A up so that you feed the test program into it, and it stops when
it establishes that the test program never halts. Now feed A into itself.
If A (test program) doesn't stop, A (predictor program) should stop and tell
you that the test program doesn't stop.
But A can't both stop and run forever. The only way out is
to assume that you can't predict which programs will halt in the first place.
Turing's argument was actually a little more complicated, but that's the
Enter the train set. In 1994, Adam Chalcraft and Michael Greene,
then undergraduates at the University of Cambridge, discovered an interpretation
of Turing machines in terms of a toy train wandering around a track. The
Turing-machine layout has a depot from which the train starts, representing
the start of a program, and a station, which represents the end of the
computation. Each memory cell of a Turing machine can be represented by a
"circuit" or network of track and points, and the contents of each memory
cell depend on the states of the points within it.
You make as many sub-layouts as you need to have enough memory
for the calculation. The layout is programmed by setting the points to particular
states. The train is set off and it wanders through the layout, switching
points as it passes through them. If it gets to the station, it stops ; the
results of the computation can then be read off from the various states of
Using this interpretation, Turing's theorem implies that no
formal theory, when presented with a randomly chosen layout, can predict
whether the train will eventually reach the station. If a test for "does
the train halt?" existed, you could construct a layout that suffers from
the same problem as the computer program A-the train reaches the station
if, and only if, it doesn't. That's obviously nonsense, so no decision procedure
for halting can exist.
Admittedly, this is a somewhat banal example -being unable
to predict the long-term motion of a toy train doesn't sound like a particularly
serious limitation. But it demonstrates that limits do exist to theoretical
science. But what, then, about practical science?
John Barrow, an astronomer at the University of Sussex
talks about several kinds of fundamental limits to practical science. One
involves technological limits that are inherently intractable. For instance,
a few years ago, theoretical physicist Rolf Landauer from IBM's research
centre at Yorktown Heights in New York asked whether answering certain questions
require more resources- in space, time or energy-than are available in the
entire Universe. Consider, for example, the state of the Universe as a whole.
In a classical
(non-quantum) model, it is possible in theory to describe the evolution
of the Universe by specifying its initial conditions-the precise state of
every particle an instant after the big bang, say. Then the equations of
physics will allow you to deduce all future states.
However, to specify the initial conditions you have to record
a list of numbers for every constituent particle. Are there enough particles
available to do this? It's a moot point. Even more problematic is the issue
of what would happen if you disturbed the motion of a large proportion of
the particles in the Universe in the mere act of recording those initial
conditions : arguably, the future you predicted would be disturbed by how
you set up the computation. It seems likely that trying to predict the behaviour
of a system as complex as the Universe, while carrying out the prediction
inside that system, is a self- defeating task.
Frontiers of knowledge
Other examples of scientific limits involve genuine no-go areas
: what you want to do sounds reasonable, and you can imagine doing it-it
just happens not to be possible. Travelling faster than light, time travel
(perhaps), or visiting a black hole and getting out again in one piece are
just not on.
These "genuine" limits have to be carefully distinguished from
what quantum physicist James Hartle calls "false limits". It is impossible
to take a holiday in Atlantis: is this a genuine limitation of air travel?
Hardly. In quantum mechanics, the Heisenberg
uncertainty principle implies that it is impossible to measure
both the position and momentum of a particle at the same time. This may look
like a genuine no-go area, but it would be fairer to interpret the uncertainty
principle as saying that a quantum particle does not possess a simultaneous
position and momentum. It's no-go because, like Atlantis, it's not there.
Another way of thinking about this is
Stephen Hawking's famous analogy about going
north of the North Pole while staying on the surface of the Earth. Once again,
it's no-go. But not because there's some physical limit stopping you getting
there. It's just a meaningless thing to try to do. This is not a limit to
scientific endeavour, so much as adopting the wrong mindset and asking the
wrong question. You're asking about something that doesn't exist.
There are other investigations that may be hampered by this
kind of "existential limit". Take the current hunt for
a Theory of
Everything that reveals the four known forces of nature (gravitational,
electromagnetic, strong and weak) as aspects of a single unified force. Though
such a theory could well exist, we can't assume that it does. The real world
might not be like that. If so, physicists will never find a way to link the
four forces no matter how hard they look. In their desire to pin down the
nature of matter, they could be looking for the wrong thing.
But once again, this is not a restriction on science. Rather
it's an indication that you need to think about the problem in a different
way. In fact, coming up against this kind of limit
can be useful, in that it can help you to realise that you are asking
the wrong question, and to work out the right one. Take
protein folding. A protein is a large
molecule (between , say, a thousand and a million atoms) composed of units
known as amino acids. Most proteins are there to manipulate other molecules
in a very specific way-for example, haemoglobin captures or releases molecules
of oxygen. But the action of the proteins depends very heavily on their exact
shape -how the amino acid chain folds up in three dimensions.
Getting a protein to fold up is no great feat, any more than
getting a piece of string to tangle. But a given chain of amino acids can,
in principle, fold up in a vast number of different ways, and the problem
is getting it to do it correctly.
A protein containing a thousand amino acids can fold itself
in about a second. Many physicists trying to model the process have worked
on the assumption that biology does this by working out the configuration
with the least energy. Unfortunately, it turns out to be incredibly difficult
to compute minimal energy configurations, even for short molecules . One
estimate quoted in the recent book Boundaries and Barriers, edited
by John Casti of the
Santa Fe Institute, is that for cytochrome c such a calculation would take
10127 years on a supercomputer.
Unlike Gödel -type limits, this one is not a limitation
in principle, it is a limitation in practice. The difficulty here is that
the number of potential configurations is vast, and the minimal-energy
configuration lurks among them like a microscopic needle inside a haystack
the size of a billion Universes. So how does this protein-or biology- perform,
in a second, a 10127 year computation? Massive parallelism? That
might get it down to 10100 years. Quantum superpositions of all
possible folding patterns, automatically generating the minimal one? Unlikely.
What's probably happening is something that biologists have
suspected for years. Biology has found a quick-and-dirty method that comes
close to minimal energy-close enough to fool literal-minded scientists into
thinking that is really what's going on. The trick may be not to start with
a complete linear chain of amino acids and then fold it up-which is what
these horrendous computations try to do. Instead, it folds the thing as it
builds it, sequentially, and that must surely reduce the computational
complexity. It also, one imagines, jiggles the part-formed protein around
every so often to prevent odd protuberances getting hooked up on extraneous
In fact, George Rose of Johns Hopkins University in Maryland,
has written a new program called LINUS that employs heuristic rules (inspired
scientific guesswork) to predict how really large proteins with a chain of
1000 amino acids will fold. It's a bit like playing
chess by using general
principles like "don't lose your queen". You play a reasonable game,
but not always the best possible one -a grandmaster might well win by breaking
the heuristic rules, say with a queen sacrifice.
LINUS works in a similar way. Instead of looking for minimal
energy, it works on principles such as "avoid shapes with energies that look
too big", and it does pretty well. Joseph Traub of Columbia University in
New York, has studied the relation between simulations of protein-folding
on computers and protein-folding as it really happens -his conclusion being
that the limitations of one need not carry over to the other.
Take it to the limit
In a way, this illustrates why we should not be afraid of
scientific limits. One of the strangest consequences of
theorem is that it had very little effect on the practice, or the growth,
of mathematics. The main reason is that there are plenty of problems left
that aren't insoluble. And anyway, extending Gödel's methods shows that
there is no way to decide in advance whether your particular problem has
a solution or not. So his theorem doesn't affect what you do: it just opens
your eyes to the possibility that you might never succeed.
There is more to it than that, of course. Knowing one's limits
is the essence of wisdom. Gödel's dramatic discovery spelt not the end
of mathematics, but its maturity, and the same goes for science. Limits are
unlikely to kill it off. Instead, they define the boundaries of what we can
study, and can help our understanding within those boundaries. Barrow also
sees limitations as a positive feature: "As we probe deeper into the intertwined
logical structures that underwrite the nature of reality, we can expect to
find more of these deep results which limit what can be known. Ultimately,
we may even find that their totality characterises the Universe more precisely
than the catalogue of those things that we can know." The biggest limitation
of science may turn out to be an inability to determine its own limitations.
After all, if science really were omnipotent, it would be able to invent
a theory so hard that scientific method couldn't come to grips with.