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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.