Mathematics: The only true universal language
16 February 2009
IF WE ever establish contact with intelligent aliens living on a planet
around a distant star, we would expect some problems communicating with
them. As we are many light years away, our signals would take many years
to reach them, so there would be no scope for snappy repartee. There
could be an IQ gap and the aliens might be built from quite different
Yet there would be much common ground too. They would be made of similar
atoms to us. They could trace their origins back to the big bang 13.7
billion years ago, and they would share with us the universe's future.
However, the surest common culture would be mathematics.
Mathematics has been the language of science for thousands of years, and
it is remarkably successful. In a famous essay, the great physicist
Eugene Wigner wrote about the "unreasonable effectiveness of
mathematics". Most of us resonate with the perplexity expressed by
Wigner, and also with Einstein's dictum that "the most incomprehensible
thing about the universe is that it is comprehensible". We marvel at the
fact that the universe is not anarchic - that atoms obey the same laws
in distant galaxies as in the lab. The aliens would, like us, be
astonished by the patterns in our shared cosmos and by the effectiveness
of mathematics in describing those patterns.
Mathematics can point the way towards new discoveries in physics too.
Most famously, British theorist Paul Dirac used pure mathematics to
formulate an equation that led to the idea of antimatter several years
before the first antiparticle was found in 1932. So will physicists'
luck hold as they aim to probe still deeper levels of structure in the
cosmos? Are limits set by the intrinsic capacity of our brains? Can
computers offer insights, rather than just crunch numbers? These are
some of the questions that exercise me.
The precedents are encouraging. The two big breakthroughs in physics in
the 20th century owed much to mathematics. The first was the formulation
of quantum theory in the 1920s, of which Dirac was one of the great
pioneers. The theory tells us that, on the atomic scale, nature is
intrinsically fuzzy. Nonetheless, atoms behave in precise mathematical
ways when they emit and absorb light, or link together to make molecules.
The other was Einstein's general relativity. More than 200 years
earlier, Isaac Newton showed that the force that makes apples fall is
the same as the gravity that holds planets in their orbits. Newton's
mathematics is good enough to fly rockets into space and steer probes
around planets, but Einstein transcended Newton. His general theory of
relativity could cope with very high speeds and strong gravity, offering
deeper insight into gravity's nature.
Yet despite his deep physical insights, Einstein was not a top-rate
mathematician. The language needed for the great conceptual advances of
20th-century physics was already in place and Einstein was lucky that
the geometrical concepts he needed had already been developed by German
mathematician Bernhard Riemann a century earlier. The cohort of young
quantum theorists led by Erwin Schrödinger, Werner Heisenberg and Dirac
were similarly fortunate in being able to apply ready-made mathematics.
Einstein was not a top-rated mathematician.
The concepts he needed had already been developed
The 21st-century counterparts of these great figures - those seeking to
mesh general relativity and quantum mechanics in a unified theory - are
not so lucky. A unified theory is key unfinished business for science today.
The most favoured theory posits that the particles that make up atoms
are all made up of tiny loops, or strings, that vibrate in a space with
10 or 11 dimensions. This string theory involves intensely complex
mathematics that certainly cannot be found on the shelf, and the
challenges it poses have been a stimulus for mathematics. Ed Witten, the
acknowledged intellectual leader of string theory, ranks as a
world-class mathematician, and several other leading mathematicians have
been attracted by the challenge.
String theory is not the only approach to a unified theory, but it is by
far the most intensively studied one. This endeavour is surely good for
mathematics, but there is controversy about how good it is for physics.
Arguments rage over whether string theory is right, whether it will ever
engage with experiment, and even whether it is physics at all. There
have even been commercially successful books rubbishin g the idea.
To me, criticisms of string theory as an intellectual enterprise seem to
be in poor taste. It is presumptuous to second-guess the judgement of
people of acknowledged brilliance who choose to devote their research
career to it. However, we should be concerned about the undue
concentration of talent in one speculative field.
Finding a unified theory would be the completion of a programme that
started with Newton. String theory, if correct, would also vindicate the
vision of Einstein and the late American physicist John Wheeler that the
world is essentially a geometrical structure.
An interesting possibility, which I think should not be dismissed, is
that a "true" fundamental theory exists, but that it may just be too
hard for human brains to grasp. A fish may be barely aware of the medium
in which it lives and swims; certainly it has no intellectual powers to
comprehend that water consists of interlinked atoms of hydrogen and
oxygen. The microstructure of empty space could, likewise, be far too
complex for unaided human brains to grasp.
String theory involves scales a billion billion times smaller than any
we can directly probe. At the other extreme, our cosmological theories
suggest that the universe is vastly more extensive than the patch we can
observe with our telescopes. It may even be infinite. The domain that
astronomers call "the universe" - the space, extending more than 10
billion light years around us and containing billions of galaxies, each
with billions of stars, billions of planets (and maybe billions of
biospheres) - could be an infinitesimal part of the totality.
There is a definite horizon to direct observations: a spherical shell
around us, such that no light from beyond it has had time to reach us
since the big bang. However, there is nothing physical about this
horizon. If you were in the middle of an ocean, it is conceivable that
the water ends just beyond your horizon - except that we know it
doesn't. Likewise, there are reasons to suspect that our universe - the
aftermath of our big bang - extends hugely further than we can see.
That is not all: our big bang may not be the only one. An idea called
eternal inflation developed largely by Andrei Linde at Stanford
University in Palo Alto, California, envisages big bangs popping off,
endlessly, in an ever-expanding substratum. Or there could be other
space-times alongside ours - all embedded in a higher-dimensional space.
Ours could be but one universe in a multiverse.
Other branches of mathematics then become relevant. We need a rigorous
language to describe the number of possible states that a universe could
possess and to compare the probability of different configurations. A
clearer concept of infinity itself is also required (see "The enigma of
The multiverse confronts us with infinities, multiplied by other
infinities - perhaps repeatedly. To bring sense to these concepts, we
must deploy the mathematics of transfinite numbers, which date back to
Georg Cantor in the 19th century. He showed that there was a rigorous
way to discuss infinity and that in a well-defined sense there are
infinities of different sizes. Without these exotic concepts,
cosmologists will not be able to firm up the concept of the multiverse
theory and decide, without paradoxes or ambiguities, what is probable
and what is improbable within it.
At its deepest level, physical reality may have a geometric intricacy
that would be satisfying to any intelligences on Earth or beyond, just
as it would have delighted the Pythagoreans. Provided we could
understand it, a unified theory that revealed this would be an
intellectual triumph. Calling it a "theory of everything", though, is
hubristic and misleading as it would offer no help to 99 per cent of
scientists. Chemistry and biology are not held up through ignorance of
subnuclear physics; still less are they dependent on the deepest
structure of space-time. String theory might unify two great scientific
frontiers, the very big and the very small, but there is a third
frontier - the very complex. That is perhaps the most challenging of
all, and it is the frontier on which most scientists work.
Finding a theory of everything would offer no help to 99 per cent
There are nonetheless reasons to hope that simple underlying rules might
govern some seemingly complex phenomena. This was intimated in 1970 by
the mathematician John Conway who invented the "game of life". Conway
wanted to devise a game that would start with a simple pattern and use
basic rules to evolve it again and again. He began experimenting with
the black and white tiles on a Go board and discovered that by adjusting
the simple rules of his game, which determine when a tile turns from
black to white and vice versa, and the starting patterns, some
arrangements produce incredibly complex results seemingly from nowhere.
Some patterns can emerge that appear to have a life of their own as they
move round the board.
The real world is similar: simple rules allow complex consequences.
While Conway only needed a pencil and paper to devise his game, it takes
a computer to fully explore the range of complexity inherent in it.
Computer simulations have given science an immense boost. And there is
no reason why computers cannot actually make discoveries, albeit in
their own distinctive way. IBM's chess-playing computer Deep Blue didn't
work out its strategy like a human player. Instead, it took advantage of
its computational speed to explore millions of alternative series of
moves and responses before deciding an optimum move. This brute force
approach overwhelmed a world champion.
The same approach could be put to good use to solve problems that have
us so far eluded us. For example, scientists are currently looking for
new superconductors that, rather than requiring low temperatures to
conduct electricity as they do now, will work at ordinary room
temperatures. This search involves a lot of trial and error, because
nobody understands exactly what makes the electrical resistance
disappear more readily in some materials than in others. Suppose that a
machine came up with a recipe for such a superconductor. While it might
have succeeded in the same way that Deep Blue defeated Russian chess
champion Garry Kasparov, rather than by having a theory or strategy, it
would have achieved something that would deserve a Nobel prize.
Simulations using ever more powerful computers will, in a similar way,
help scientists to understand processes that we neither study in our
laboratories nor observe directly. In my own subject of astronomy,
researchers can already create a virtual universe in a computer and
carry out experiments in it, such as calculating how stars form and die.
Some day, perhaps, my biological colleagues will be using them to
simulate many processes including the chemical complexities within
living cells, how combinations of genes encode the intricate chemistry
of a cell, and the morphology of limbs and eyes. Perhaps they will be
able to simulate the conditions that led to the first life, and even
other forms of life that could, in principle, exist.
However there is a long way to go before real machine intelligence is
achieved. A powerful computer can be a world chess champion, but not
even the most advanced robot can recognise and move the pieces on a real
chessboard as adeptly as a five-year-old child.
Maybe in the far future, though, post-human intelligence will develop
hypercomputers with the processing power to simulate living things -
even entire worlds. Perhaps advanced beings could even simulate a
"universe" that goes far beyond mere patterns on a chequer-board and the
best movie special effects. Their simulated universe could be as complex
as the one we perceive ourselves to be in. This raises a disconcerting
thought: perhaps that is what our universe really is.
It is fascinating to speculate whether hyper-intelligent aliens already
exist in some remote part of our cosmos. If so, would their brains
"package" reality in a mathematical language that would be
comprehensible to us or our descendents?
Martin Rees is professor of cosmology and astrophysics and master of
Trinity College at the University of Cambridge. He was appointed
Astronomer Royal in 1995 and is President of the Royal Society. This
article is based on contributions to a discussion by a panel that
included mathematicians Michael Atiyah and Alain Connes about the
relationship between mathematics and science
The enigma of infinity
Infinity is an ancient mystery and unendingness is hard to conceive. As
far back as 350 BC Greek philosophers speculated about what would happen
if you could throw a spear from the edge of space, should such a place
It seems absurd that there should be no "beyond" where it can go. Space
may curve up on itself - and so be finite but unbounded - but equally it
can go on for ever.
Infinity is qualitatively different from even the largest number. Finite
numbers, however large, obey the laws of arithmetic. You can add,
multiply and divide them, and put different numbers unambiguously in
order of size. But an infinity is the same as a part of itself, and when
it is multiplied by another number (even another infinity) it is in a
well-defined sense just the same.
A metaphor for this is known as Hilbert's hotel. Suppose a hotel is full
and each guest wants to bring a colleague who would need another room.
This would be a nightmare for the management, who could not double the
size of the hotel instantly.
In an infinite hotel, though, there is no problem. The guest from room 1
goes into room 2, the guest in room 2 into room 4, and so on. All the
odd-numbered rooms are then free for new guests.