Einstein's Genius Club by Feldman Burton Williams Katherine

Author:Feldman, Burton, Williams, Katherine
Language: nld
Format: epub
Publisher: Skyhorse Publishing, Inc.
Published: 2011-11-17T16:00:00+00:00

PART 3

THE UNIVERSE

Physics, mathematics, and the universe—these three words form the angles of a tangled and intimate set of relations. Einstein and Pauli the physicists, Gödel and Russell, the mathematicians—each worked within a science that attempts to describe the actual world. That, at least, was the purpose of mathematics at its inception (Euclid's geometry) and the effect of physics in the nineteenth and early twentieth century.

THE LOGIC OF PARADOX

BEFORE WE TURN TO RELATIVITY, quantum mechanics, and the search for a unified theory, we shall take a brief detour into another world altogether—that of mathematical logic. Like physics, the world of mathematics underwent revolutionary changes throughout the nineteenth century and into the twentieth. In so doing, it virtually merged with the doctrines of analytical philosophy and logicism. Few players in the twin worlds of mathematics and logic were more influential than Russell and Gödel.

There is good reason for starting with mathematics. True, physics began from observations of the visible world. Yet it evolved through mathematics. From the late nineteenth century on, mathematics became an essential tool of the physicist. Though mathematics was never absent from early modern physics—Newton invented calculus, after all—in the twentieth century, mathematics overtook empiricism as the primary method for generating physics. What Newton could observe (albeit through eyes made keen by the imagination) in a falling apple or a setting moon no longer mattered in twentieth-century physics.

Mathematics does not describe the physical world per se. It does, however, problem-solve in the realms of space, number, form, and change. Through mathematics, Einstein explored four-dimensional geometries never seen on land or sea. Today, the mathematics of string theory yields nine space dimensions. These are not observable phenomena. The nine space dimensions cannot even be explained properly in nonmathematical terms.1 We are led to these proposals not through observation, but through mathematics. Einstein, a born physicist schooled in nineteenth-century empiricism, approached mathematical formalism with trepidation: “As far as the laws of mathematics refer to reality, they are not certain; as far as they are certain, they do not refer to reality.”2

For pragmatic physicists, mathematical formalism either works or not. Mathematics is a tool. Why does it work? No one has a satisfactory answer. In his celebrated paper, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” the physicist Eugene Wigner pondered the seeming miracle of the mathematics-physics connection:

The mathematical formulation of the physicist's often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena. This shows that the mathematical language has more to commend it than being the only language which we can speak; it shows that it is, in a very real sense, the correct language.3

Mathematics, like experimentation, sometimes yields surprising or even unwanted results, as if it, too, were beyond human control. In 1928, the British physicist Paul Dirac formulated an equation only to find that it predicted a hitherto unknown and startling particle, the antielectron (or positron). One might even say that it was not Dirac, but his equation (via a minus sign), that discovered anti-matter.

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