New York Times, February 10, 1998, Tuesday
SCIENCE DESK
AT the top of the list of science's unanswered
questions, like what is consciousness and how did life begin, is the deepest
mystery of all: Why does the universe appear to follow mathematical laws?
According to the Big Bang theory,
matter, energy, space and time were created during the primeval explosion.
Instantly, it seems, everything began unfolding according to a mathematical plan.
But where did the mathematics come from? What are the origins of numbers and
the relationships they obey?
The ancient followers of the Greek
mathematician Pythagoras declared that numbers were the basic elements of the
universe. Ever since, scientists have embraced a kind of mathematical
creationism: God is a great mathematician, who declared, ''Let there be
numbers!'' before getting around to ''let there be light!''
Scientists usually use the notion
of God metaphorically. But ultimately, most of them at least tacitly embrace
the philosophy of Plato, who proposed, rather unscientifically, that numbers
and mathematical laws are ethereal ideals, existing outside of space and time
in a realm beyond the reach of humankind.
Because the whole point of science
is to describe the universe without invoking the supernatural, the failure to
explain rationally the ''unreasonable effectiveness of mathematics,'' as the
physicist Eugene Wigner once put it, is something of a scandal, an enormous gap
in human understanding.
''We refuse to face this
embarrassment,'' Reuben Hersh, a mathematician emeritus of the University of
New Mexico in Albuquerque, wrote in his recent book, ''What Is Mathematics
Really?'' (Oxford University Press, 1997). ''Ideal entities independent of
human consciousness violate the empiricism of modern science.'' While science
is anchored in observations of the physical world, Dr. Hersh insists that
mathematics is more of a human creation, like literature, religion or banking.
Dr. Hersh's book is one of several
recent works contending that mathematics is not an ethereal essence but comes
from people who invented, not discovered it. The sentiments presented in the
books are not entirely new and the mathematical puzzle has hardly been solved.
But the idea of a human-centered mathematics may be gaining force and respect.
In ''The Number Sense: How the Mind
Creates Mathematics'' (Oxford University Press, 1997), Stanislas Dehaene, a
cognitive scientist at the National Institute of Health and Medical Research in
Paris, marshals experimental evidence to show that the brains of humans -- and
even of chimpanzees and rats -- may come equipped at birth with an innate,
wired-in aptitude for mathematics. Gregory J. Chaitin, a mathematician at
I.B.M.'s Thomas J. Watson Research Center in Yorktown Heights, N.Y., takes an
anti-Platonist stance in ''The Limits of Mathematics'' (Springer, 1997). Two
Berkeley scientists, George Lakoff and Rafael E. Nunez, are working on a book
tentatively called ''The Mathematical Body,'' contending that even the most
abstract mathematical concepts arise from basic human experience -- from the
way the body interacts with the world. They gave a preview of their ideas in a
chapter of another book published last year: ''Mathematical Reasoning:
Analogies, Metaphors and Images,'' edited by Lyn D. English (Erlbaum).
The authors are all working
mathematicians and scientists, not postmodern critics viewing the territory
from afar. They emphatically reject those who try to dismiss mathematics and
science as arbitrary constructions, or white male Eurocentric folklore. But
they are just as adamant in rejecting what most mathematicians and many
scientists have come to take for granted: the Platonic creed.
''The normal notion of pure math is
that mathematicians have some kind of direct pipeline to God's thoughts, to
absolute truth,'' Dr. Chaitin wrote in ''The Limits of Mathematics.'' While
scientific knowledge is tentative and subject to constant revision, mathematics
is usually seen as eternal. But Dr. Chaitin called on his colleagues to abandon
mathematical Platonism and adopt a ''quasi-empirical'' approach that treats
mathematics as just another messy experimental science.
''Quasi-empirical,'' he said,
''means that math ain't that different from physics.'' This view is laid out in
detail in a revised edition of ''New Directions in the Philosophy of
Mathematics,'' edited by Thomas Tymoczco (Princeton University Press, 1998).
Leopold Kronecker, a 19th-century
mathematician, once said: ''The integers were created by God; all else is the
work of man.'' Albert Einstein, taking a different view of whole numbers, wrote
that ''the series of integers is obviously an invention of the human mind, a
self-created tool which simplifies the ordering of certain sensory
experiences.''
In ''The Number Sense,'' Dr.
Dehaene went even further. The integers -- the smallest ones, anyway -- are
hard-wired into human nervous systems by evolution, along with a crude ability
to add and subtract. Mathematics, he believes, is ''engraved in the very
architecture of our brains.''
''Because we live in a world full
of discrete and movable objects, it is very useful for us to be able to extract
number,'' he argued in a recent forum published on the Internet (www.edge.org)
by the Edge Foundation. ''This can help us to track predators or to select the
best foraging grounds, to mention only very obvious examples.''
By studying brain-damaged patients
who have lost basic number skills, Dr. Dehaene and others have tentatively
traced this arithmetical module to an area of the brain called the inferior
parietal cortex, a poorly understood location where visual, auditory and
tactile signals converge. Scientists are intrigued by clues that this region is
also involved in language processing and in distinguishing right from left.
Mathematics is, after all, a kind of language intimately involved with using
numbers to order space. The inferior parietal cortex also seems to be important
for manual dexterity, and arithmetic begins with counting on the hands. Imaging
experiments, in which people's brains are monitored as they calculate, point to
the same region as a primitive number processor.
If this neurological calculator has
indeed been bequeathed by evolution, then traces of it should be found in other
species. In making his argument, Dr. Dehaene draws on experiments over the last
few decades suggesting that even rats have a rudimentary number sense. The
animals were taught to press lever A four times and then lever B to get food,
or to press lever A when they heard a two-tone sequence and lever B when they
heard an eight-tone sequence. (To insure that the rats were responding to the
number of signals and not just to their duration, the two-tone sequence
sometimes lasted longer than the eight-tone one.)
Even more striking were later
experiments in which rats were first trained to associate lever A with two
tones and lever B with four tones. Then they were taught to associate A with
two flashes of light and B with four flashes. If the rats heard two tones and
saw two flashes they learned to push B, not A. They seemed to have comprehended
the notion that two plus two equals four.
The rats were not precise. Trained
to press one lever four times, they often pressed it five or six times,
expecting to be rewarded just the same, or they confused a seven-tone sequence
with an eight-tone one. But the experiments support the notion of a primitive
neurological number processor, even in rodents.
In other experiments, chimpanzees
seemed to learn simple arithmetic. Given a choice between one tray with a pile
of three chocolate chips and another pile of four and a second tray with piles
of two and three chips, they chose the first tray with the most candy. But when
the totals on the trays differed by only one chip, the chimps were less likely
to make the discrimination. The number sense is approximate, not exact. More
recent experiments on infants, using Mickey Mouse toys instead of chocolate
chips, found signs of the same kind of rough numerical ability in babies less
than 5 months old.
Dr. Dehaene says this instinct is
innate, as singing is for songbirds or spinning webs is for spiders. Numbers
are not Platonic ideals but neurological creations, artifacts of the way the
brain parses the world. In that sense they are like colors. Red apples are not
inherently red. They reflect light at wavelengths that the brain, as it was
wired by evolution, interprets as red.
While people are born with an
understanding of the rudiments of arithmetic, he contends, going beyond that
requires learning and creativity. Multiplication, division and the whole
superstructure of higher mathematics -- from algebra and trigonometry, to
calculus, fractal geometry and beyond -- are a beautiful improvisation, the
work of human culture.
The ability to weave simple ideas,
like two plus two equals four, into the tapestries of higher mathematics, he
suggests, is not unlike the human skill for language. People take a relatively
small collection of words and, using a few simple rules of grammar and syntax,
create literature.
At the University of California at
Berkeley, Dr. Lakoff, a linguist and cognitive scientist, and Dr. Nunez, a
developmental psychologist, contend that the source of mathematics lies not
just in the brain but in the human body and the physical world. People favor
number systems based on 10 because they have 10 fingers and 10 toes. But that
is just the beginning of the story.
Driven by a built-in number sense,
the theory goes, primitive people explored the wonders of counting by playing
with their fingers or putting rocks in a pile. But they found that counting
could also be thought of as taking steps along a line to measure distance. That
metaphor eventually allowed for the invention of more abstract concepts. Walk
one way and you get the positive integers; walk the other way and you get the
negative integers. The starting point is zero.
Multiplication by a positive number
can be thought of as stretching; multiplying by a negative number makes
something shrink.
Dr. Lakoff and Dr. Nunez call these
''grounding metaphors.'' In inventing mathematics, they contend, people also
used ''linking metaphors'' to connect two sets of ideas. The sequence of
numbers can be mapped onto the notion of a line. Now numbers are not fingers or
rocks but points. Put two lines together at right angles and you get what
mathematicians call a Cartesian plane, a two-dimensional graph that opens up a
whole new arena to play in.
And so, floor by floor, the tower
of mathematics is built. ''Students never learn that mathematics is a creative
endeavor,'' Dr. Lakoff said in a recent interview. ''Mathematics is more
glorious because it is humanly constructed.'' There is no such thing as pure
mathematics or pure thought, he said -- they are physical activities.
That does not mean that mathematics
is a relativistic free-for-all. The most basic mathematical inventions are
rooted in the brain and body. Even mathematicians' loftier elaborations are
tested against the universe. Of the infinite range of mathematical creations,
scientists keep those that help them explain and predict reality.
Mathematicians savor the others as ends in themselves, like paintings or
symphonies.
But many scientists and
mathematicians still doubt that evolution -- biological or cultural -- can
adequately explain why mathematics works so well in describing the fundamental
laws of the universe.
''Our ability to discover, and
describe mathematically, Newton's equations has no immediate survival value,''
said Dr. Paul Davies, professor of mathematical physics at the University of
Adelaide in Australia. ''This point has even greater force when it comes to,
say, quantum mechanics. The reason people find it hard to understand quantum
physics is precisely because there is no survival value in being able to do
so.''
The reason mathematics is so
effective, he says, remains a deep mystery. ''No feature of this uncanny
'tuning' of the human mind to the workings of nature is more striking than
mathematics,'' he wrote in ''The Mind of God: The Scientific Basis for a
Rational World'' (Simon & Schuster, 1992).
Some hold out vague hopes that the
mystery might be solved if humans ever encounter an alien civilization. If
mathematics is indeed universal and eternal, the theory goes, then the aliens
would understand concepts like pi, the ratio of a circle's circumference to its
diameter. The Platonists' assume that there is ''pi in the sky,'' as the
British astronomer John D. Barrow said in a book by that name (Oxford
University Press, 1992).
The anti-Platonists say there is no
reason to believe the aliens would understand mathematical inventions from
Earth. ''The Platonist claim that every intelligence must produce prime
numbers, pi and the continuum hypothesis is an example of simple
anthropomorphism,'' Dr. Hersh said.
But if earthlings were utterly
baffled by extraterrestrial mathematics, would the anti-Platonists have proved
their point? Not necessarily.
''Alien intelligences may be so far
advanced that their math would simply be too hard for us to grasp,'' Dr. Davies
said. ''The calculus would have baffled Pythagoras, but with suitable tuition
he would have accepted it.''
But what if the humans and the
aliens could communicate mathematically? Would that decide the issue in favor
of the Platonists? Not really.
''If the alien species had evolved
in an environment similar to ours -- say, a world composed of distinct, movable
objects -- then most likely its brain would have incorporated, through natural
selection, the same regularities about the external world as we have,'' Dr.
Dehaene said. ''Thus, it would have a very similar arithmetic and geometry.
''But now, suppose that the alien
species has evolved in a radically different environment, like a fluid world,''
he continued. ''Then knowledge of movable objects would not be essential to its
survival, while knowledge of fluid mechanics, vortices, etc. would be. I
believe that this hypothetical species would have internalized in its brain
regularities strikingly different from ours. Hence it would have radically
different mathematics.''
And so the argument continues to
churn.
Several years ago, the French
mathematician Alain Connes, arguing for the Platonists, and the French
neurobiologist Jean-Pierre Changeux, taking the opposite side, tried to settle
the matter with a debate. The result, translated and edited by M. B. DeBevoise,
was the book ''Conversations on Mind, Matter and Mathematics'' (Princeton
University Press, 1995).
Ranging over a vast field of topics
including relativity, quantum mechanics, neurobiology, topology, game theory,
information theory and non-Euclidean geometry, the two reached the end of their
discussion with no resolution.
The
best they could do was to agree to disagree.
Copyright 2002 The New
York Times Company