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Saturday, July 25, 2015

Math Works Great—Until You Try to Map It Onto the World

"In 1900, the great mathematician David Hilbert presented a list of 23 unsolved problems worth investigating in the new century." according to Natalie Wolchover, staff writer at Quanta Magazine. 

Marshall Slemrod of the University of Wisconsin, Madison, lecturing at the University of Leicester in August 2014. Courtesy of Marshall Slemrod.

The list became a road map for the field, guiding mathematicians through unexplored regions of the mathematical universe as they ticked off problems one by one. But one of the problems was not like the others. It required connecting the mathematical universe to the real one. 
Hilbert’s sixth problem called upon researchers to axiomatize the laws of physics—that is, rigorously construct them from a basic set of starting assumptions, or axioms. Doing so would reveal contradictions between laws that demanded different axioms. And deriving the entire body of physical laws from the same axioms would prove they were not merely haphazard, incoherent descriptions of disparate phenomena, but instead formed a unified, mathematically airtight, internally consistent theory of reality. “Once again it was an issue of unification, which pervades physics to this day,” said Marshall Slemrod, a mathematician at the University of Wisconsin, Madison.

Axiomatizing all of physics was a tall order, so Hilbert proposed a specific task: Determine whether the microscopic and macroscopic pictures of a gas rest on equivalent axiomatic foundations, and are thus different manifestations of a single theory. Experts approached this problem by attempting to mathematically translate the Boltzmann equation, which describes a gas as microscopic particles bouncing around at a range of speeds, into the Navier-Stokes equations, which describe the gas on larger scales as a continuous, flowing entity. Could the particle and fluid pictures be rigorously linked?

While Hilbert’s broader aim of axiomatizing physics remains unfulfilled, recent research has yielded an unexpected answer to the particle-fluid question. The Boltzmann equation does not translate into the Navier-Stokes equations in all cases, because the Navier-Stokes equations—despite being exceptionally useful for modeling the weather, ocean currents, pipes, cars, airplane wings and other hydrodynamic systems, and despite the million-dollar prize offered for their exact solutions—are incomplete. The evidence suggests that truer equations of fluid dynamics can be found in a little-known, relatively unheralded theory developed by the Dutch mathematician and physicist Diederik Korteweg in the early 1900s. And yet, for some gases, even the Korteweg equations fall short, and there is no fluid picture at all.

“Navier-Stokes makes very good predictions for the air in the room,” said Slemrod, who presented the evidence last month in the journal Mathematical Modelling of Natural Phenomena. But at high altitudes, and in other near-vacuum situations, “the equations become less and less accurate.”...

Leo Corry, a historian of mathematics at Tel Aviv University in Israel who has written a book about David Hilbert and his sixth problem, notes that Hilbert’s original aim seems to have gotten lost in the details of the particle-fluid question and remains unaddressed. “Notice that the words ‘axiom’ or even ‘foundation’ or ‘conceptual analysis’ do not appear even once in Slemrod’s review,” Corry said.
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Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.

Additional resources
Hilbert's sixth problem (Wikipedia, the free encyclopedia)

Source: Wired