[Paleopsych] LAT: Definitional Drift: Math Goes Postmodern
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Definitional Drift: Math Goes Postmodern
http://www.latimes.com/news/opinion/commentary/la-oe-wertheim16may16,0,2648360,print.story
By Margaret Wertheim
Margaret Wertheim is the "Quark Soup" science columnist for LA Weekly
and is working on a book about the role of imagination in theoretical
physics.
May 16, 2005
A baker knows when a loaf of bread is done and a builder knows when a
house is finished. Yogi Berra told us "it ain't over till it's over,"
which implies that at some point it is over. But in mathematics things
aren't so simple. Increasingly, mathematicians are confronting
problems wherein it is not clear whether it will ever be over.
People are now claiming proofs for two of the most famous problems in
mathematics -- the Riemann Hypothesis and the Poincare Conjecture --
yet it is far from easy to tell whether either claim is valid. In the
first case the purported proof is so long and the mathematics so
obscure no one wants to spend the time checking through its hundreds
of pages for fear they may be wasting their time. In the second case,
a small army of experts has spent the last two years poring over the
equations and still doesn't know whether they add up.
In popular conception, mathematics is the ultimate resolvable
discipline, immune to the epistemological murkiness that so bedevils
other fields of knowledge in this relativistic age. Yet Philip Davis,
emeritus professor of mathematics at Brown University, has pointed out
recently that mathematics also is "a multi-semiotic enterprise" prone
to ambiguity and definitional drift.
Earlier this year, Davis gave a lecture to the mathematics department
at USC titled "How Do We Know When a Problem Is Solved?" Often, he
told the audience, we cannot tell, for "the formulation and solution
of problems change throughout history, throughout our own lifetimes,
and even through our rereadings of texts."
Part of the difficulty resides in the notion of what we mean by a
solution, or as Davis put it: "What kind of answer will you accept?"
Take, for instance, the task of trying to determine whether a very
large number is prime -- that is, it cannot be split evenly into the
product of any smaller components, except 1. (Six is the product of 2
by 3, so it is not prime; 7 has no smaller factors, so it is.)
Determining primeness has huge practical consequences -- prime numbers
are widely used in computer security codes, for instance -- yet when
the number is large it can take an astronomical amount of computer
time to determine its primeness unequivocally. Mathematicians have
invented statistical methods that will give a probabilistic answer
that will tell you, for instance, a given number is 99.99% certain to
be prime. Is it a solution? Davis asked.
Other problems can also be addressed by brute computational force, but
many mathematicians feel intrinsically uncomfortable with this
approach. Said Davis: "It is certainly not seen as an aesthetic
solution." A case in point is the four-color map theorem, which
famously asserts that any map can be colored with just four colors (no
two adjoining sections may be the same color).
The problem was first stated in 1853 and over the years a number of
proofs have been given, all of which turned out to be wrong. In 1976,
two mathematicians programmed a computer to exhaustively examine all
the possible cases, determining that each case does indeed hold. Many
mathematicians, however, have refused to accept this solution because
it cannot be verified by hand. In 1996, another group came up with a
different (more checkable) computer-assisted proof, and in December
this new proof was verified by yet another program. Still, there are
skeptics who hanker after a fully human proof.
Both the Poincare Conjecture (which seeks to explain the geometry of
three-dimensional spheres) and the Riemann Hypothesis (which deals
with prime numbers) are among seven leading problems chosen by the
Clay Mathematics Institute for million-dollar prizes. The institute
has its own rules for determining whether any one of these problems
has been solved and hence whether the prize should be awarded.
Critically, the decision is made by a committee, which, Davis said,
"comes close to the assertion that mathematics is a socially
constructed enterprise."
Another of the institute's million-dollar problems is to find
solutions to the Navier-Stokes equations that describe the flow of
fluids. Because these equations are involved in aerodynamic drag they
have immense importance to the aerospace and automotive industries.
Yacht designers must also wrestle with these legendarily difficult
equations. Over lunch, Davis told a story about yacht racing. He had
recently talked to an applied mathematician who helped design a yacht
that won the America's Cup. This yachtsman couldn't have cared less if
the Navier-Stokes equations were solved; what mattered to him was
that, practically speaking, he could model the equations on his
computer and predict how water would flow around his hull. "Proofs,"
said Davis, "are just one of the tools that mathematicians now use."
We may never fully solve the Navier-Stokes equations, but according to
Davis it will not matter. Like so many other fields, mathematics is
becoming less about some Platonic ideal of ultimate answers, and more
a functional project of computational simulation and communal
negotiation. Dare we say it: Math is becoming postmodern.
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