**Useful
Invention Or Absolute Truth: What Is Math?**

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

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

''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

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,

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.

*By George Jonhson, February 10th 1998*