Forget the likes of Terminator and Wall-E - the first intelligent
robot to stalk this earth could be seriously square, says Michael
Brooks
by Michael Brooks
IN December, philosopher and artificial intelligence expert Aaron
Sloman announced his intention to create nothing less than a robot
mathematician. He reckons he has identified a key component of how
humans develop mathematical talent. If he's right, it should be
possible to program a machine to be as good as us at mathematics, and
possibly better.
This is no mad quest, insists Sloman, of the University of Birmingham
in the UK. "Human brains don't work by magic, so whatever it is they
do should be doable in suitably designed machines," he says.
Human brains don't work by magic, so whatever it is they do should be
doable by machine
Sloman's creature is not meant to be a mathematical genius capable of
advancing the frontiers of mathematical knowledge: his primary aim,
outlined in the journal Artifical Intelligence (vol 172, p2015), is
to use such a machine to improve our understanding of where our
mathematical ability comes from. Nevertheless, it is possible that
such a robot could take us beyond what mathematicians have achieved so
far. Forget robot vacuum cleaners and android waitresses; we're
talking about a machine that could spawn a race of cyber-nerds capable
of creating entirely new forms of mathematics.
The field of artificial intelligence has promised much before, of
course. Early researchers thought it might open a fast-track to
understanding consciousness, and there were claims that artificially
intelligent computers and robots would change the world. The truth has
been more prosaic. AI has done some clever things, such as give us
great chess players and voice recognition software, but it hasn't
delivered a revolution.
But when it comes to mathematics, we can't rule one out yet, says
Alison Pease, who researches the philosophy of mathematics at the
University of Edinburgh, UK. Pease teaches computers to do mathematics
using AI programs, and thinks a computer really could astonish its
programmer with a new mathematical insight. "Ours hasn't yet, but
there is no reason why one shouldn't in the future," she says.
The first concrete step towards this scenario came with a program
written by Simon Colton, now at Imperial College London. The program
was named HR, in honour of the mathematicians Godfrey Harold Hardy
and Srinivasa Ramanujan. It looked for "interesting" sequences of
numbers (New Scientist, 24 February 2001, p 13).
Some of HR's discoveries have even been published - and HR, rather
than Colton, got the credit. Though they might not look like
cutting-edge advances, they could yet prove important. "I always refer
to HR's work in number theory as recreational mathematics, but things
that look insignificant can end up being hugely significant and
interesting," Colton says.
Pease and her colleagues Alan Smaille and Markus Guhe have recently
taken things further. In their Edinburgh computing laboratory they
have been running virtual mathematics conferences, populated entirely
by digital mathematicians (see "Reinventing the conjecture"). So
where might that lead?
All the way to significant new mathematics, Sloman hopes. His idea is
that our key mathematical capabilities are formed in childhood. So
rather than engineering a fully fledged mathematician's brain, Sloman
thinks we should build a robot with a child-like brain and let it grow
into its mathematical destiny.
There's just one problem. How do we know which of our childhood
capabilities equip us for a life of juggling numbers?
Sloman is busy gathering clues. The answer, he reckons, lies in the
spatial awareness skills that children must acquire in order to
negotiate their world: skills such as knowing that a toy train pushed
into a tunnel will come out the other side. Or that a jigsaw puzzle
piece fits its gap only when correctly oriented. Or that the number of
toys on the sofa does not depend on the order in which you count them.
>From the minds of babes
You might be surprised to learn, for instance, that you grasped the
topological concept called "the transitivity of containment" when you
were still a toddler. Stacking cups, one inside the other, you learned
that the small cup would fit not only in the medium-sized cup, but
also inside the big one.
Transitivity of containment, like other geometrical and topological
concepts, is learned through experience. "There are hundreds, if not
thousands more examples of things a child learns empirically, that are
later seen to be theorems in topology, geometry and arithmetic,"
Sloman says.
At some point, children make that jump for themselves. As toddlers, we
soon translate our experiences into general theorems which we use to
make predictions.
Take the train-through-a-tunnel example. By repeated experiences like
this, toddlers learn the basic properties of rigid rods. That's why a
3-year-old carrying a long broom handle can negotiate a narrow
corridor, turn a corner at the end without getting the broom handle
caught in the vertical bars of a stair-gate, then make adjustments so
that the handle will go through the next doorway. "There is a switch
from learning empirically to realising it has 'simply got to be like
that'," Sloman says.
And here is the key to the emergence of the mathematical mind. "The
mechanisms that make that possible in a child are related to what
makes it possible for them to go on to become a mathematician," Sloman
says. "A lot of abstract maths has its roots in our ability to think
about space and time, processes, and interactions between processes
and structures."
Sloman has gone back to basics, to watch how children learn to
navigate the world around them. He is building an archive of
observations of children performing pseudo-mathematical tasks. These
navigational and object manipulation skills - or at least the ability
to acquire them quickly - must be encoded in the genome, Sloman
reckons. And that means they could be encoded in a machine.
Sloman is still a long way from designing his robot toddler. Once he
has catalogued the abilities of children at various stages of
development, he still has to work out how to understand the
mathematical implications of those abilities, then represent them in
some form of computer code. "Information needs to be encoded in some
form in order to be usable," he says. The gargantuan scale of the task
means his aims are necessarily modest: at this stage he is simply
trying to show a link between spatial manipulations and the basics of
mathematics. Anything more would be a bonus. But just how big could
that bonus be? Could a robot mathematician really do something
interesting?
"In principle, yes, absolutely," Pease says. But, she adds, the
story-so-far tempers her optimism. "Of all the scientific and
mathematical discovery programs I've looked at, nothing has yet made a
big discovery." At the very least, she says, that means there is a
long way to go.
Colton thinks there is every reason to believe computers could produce
something interesting to mathematicians. "Software is already
producing theorems of value to maths," he points out. "Not of huge
value, I admit - but then the average student or mathematician isn't
producing anything of huge value either."
He and his team are convinced that computers can be genuinely
creative. "Creativity is a very loaded word: people like to think it's
a uniquely human attribute," he says. "The fact is, computers doing
maths are more likely to be creative than, say, an undergraduate
student, in many ways."
Others are sceptical of this view. Computers are a useful tool, says
Rafael Nunez an expert on mathematical cognition at the University
of California, San Diego, but the sense that computers can invent
mathematics is an illusion. Though it looks like we can make progress
by programming machines to do mathematics, he reckons there can be
nothing in these machines that isn't pre-ordained by human
mathematical concepts. "For me, it's like computing the decimal places
of pi," Nunez says. "Once we have decided what the right rules are,
we're just using the computer to crunch numbers."
Sloman thinks Nunez's view is too narrow. He points to "evolutionary
algorithms" as a reason for optimism. This innovation allows a
computer to evolve its own programs by producing lots of them, testing
them against a goal criteria, and then selecting and "interbreeding"
the best ones. It has allowed computers do things that nobody
programmed them to do. "In some cases no human even knows how they do
what they do," Sloman says. Aerospace and automobile designers have
been using evolutionary algorithms since the late 1980s to optimise
aircraft parts and streamline their designs. Even city traders are
using them to buy and sell shares (New Scientist, 28 July 2007, p
26).
Evolution has a few million years head start on us in developing
brilliant mathematicians, of course, but at least we're now in the
race. "Our big discovery would be how do we do mathematics, rather
than how do we write a program that can generate really new
mathematics," says Pease. "But hopefully one would lead on from the
other."
Reinventing the conjecture
The traditional view of mathematics sees it as a set of some eternally
existing rules that describe the universe. Doing maths involves
exploring this abstract, ethereal domain.
Though appealing to many, this notion of mathematicians as intrepid
explorers is nothing more than a romantic myth, according to Alison
Pease of the University of Edinburgh, UK. "Maths is not discovery,"
she says. "It's a thing that we invent."
It is something that her computers can invent too, she insists. Pease
runs an AI program called HRL, which puts together "agents" in a
student-teacher relationship.
The students are programmed to take some input information, make
inferences from it and try to assess just how "interesting" those
inferences are. If sufficiently interesting, the teacher gets
involved, calling a group brainstorm designed to develop the ideas
further.
One of HRL's early successes was the independent invention of a
mathematical proposition called Goldbach's conjecture. One of the
students was given the concept of integers and divisors, and
instructed to use these to play around with the integers 1 to 10,
looking for interesting relationships. A second student had the same
concepts and instructions, but played with the integers 11 to 20.
Student two generated two new concepts: "even numbers" and "the sum of
two primes". Then it generated a conjecture: that all even numbers can
be expressed as the sum of two primes. It thought this was
interesting, and sent its work to the teacher to be placed on the
agenda for discussion.
The response was positive. "The teacher sent a request for
modifications to this conjecture, and student one found the
counterexample," Pease says. That counterexample is the number 2: the
conjecture was modified to "all even numbers except 2 are the sum of
two primes".
The fact that Christian Goldbach came up with this still unproven
conjecture in 1742 makes it a little less impressive, but the point is
made. Even if computers are a few centuries behind, it seems that
machines really can do what human mathematicians do.
Michael Brooks is writer based in Lewes, UK, and author of 13 Things
That Don't Make Sense (Profile)
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