Seeking Insights_On Mathematics, RH, and Braque

January 2013

 

Mathematics, like most human creative endeavors, is seeking and communicating insights with mere and passing interest in being clever. This is why proofs are rewritten, and some many times over, until they are distilled to a few sentences that capture their essence.  Artists likewise rework their canvases.  And, as insights arise, often to the point that a painting may be revealed to contain an entirely different composition beneath its final surface.

One dramatic example of mathematical proof rewriting is Norbert Wiener's 1932 Tauberian theorem, for which the original proof conjured sixty pages of painfully difficult analysis showing how well-known summability theory results from a field of Tauberian theorems are captured in the principles of Harmonic Analysis. Having contributed the theory of Banach algebras, Israel Moiseevich Gelfand (1913-2009) could reinterpret the essence of Wiener's theorem and numerous corollaries using his own methods. In 1941, Gelfand provided an elegant proof [1., 1a.] of Wiener's 1/f theorem: Suppose f(x) on the d-torus has an absolutely convergent Fourier Series and f(x) is nonzero on the d-torus. Then the function g(x) = 1/ f(x) also has an absolutely convergent Fourier Series. Of Gelfand his mathematical descendent Edward Frenkel said: “He was sort of an oracle who had a keen ear for the beauty of mathematics. He could see its interesting and promising directions, and help you feel the unity of mathematics.” [2.]

Another example of mathematical rewriting is the proof of Fermat's theorem on sums of two squares by Don Zagier (1951- ) which was published in the February 1990 issue of the American Mathematical Monthy [3.]. Zagier's proof consists in a single sentence. This one-line proof took 350 years of labor, insights, and new mathematics to coalesce. It all began in 1640 when Pierre de Fermat (1601?-1665) stated that an odd prime number p can be expressed as
p = x² + y², with integers x and y, if and only if (p - 1) is divisible by 4. Characteristically, Fermat offered no proof of his theorem, electing instead to torment contemporaries and fellow mathematicians for centuries to come. This was long before David Hilbert, mathematical formalism, or ArXiv, and one accepted way to go about doing mathematics—merely noting certain insights about "the queen of sciences" (Gauss) in the margin of books or some other paper then suspiciously kept hidden for decades.

Proving mathematical conjectures can be an enterprise of significant magnitude. The proof of the theorem on the sum of two squares is one instance among a great many indicating that the gist of mathematics is not in the calculations. It is not calculations that are published in mathematics research journals, it is new mathematical insights that are. Yet it is not unusual for today's mathematicians to be seduced by computational puzzles and short theorems that entail computer driven, awful, and frighteningly long proofs. This is the sort of problems that mathematical intuition has long been trained to avoid in favor of sound structure, as it is theorems with short, well understood proofs, that are actually useful in the passing on and the design of future mathematics. Like good architectural design, good mathematics take into account the people who will be working with and moving within the system. Good theorems make for good design principles.

Fermat's Little Theorem and Wiener's Tauberian theorem (named after the founder of cybernetics) have each played their role in the formulation and investigation of a long standing unproven idea in mathematics, the Riemann Hypothesis.  For an investigator of extraordinary creativity, the history of the search for a proof of the Riemann Hypothesis is packed full of examples bringing together such diverse elements as the properties of prime numbers, the sound of a vibrating violin string, quantum physics, and chaos theory. Furthermore, should RH not be correct, mathematicians feel that the human intuition regarding mathematics could never again be trusted. RH has to do with prime numbers, of which mathematicians have a sense that they are not only part of a scaffold that informs mathematics but also part of the natural world. The harmonic series, which dates back to Pythagoras, played a central role in the formulating of RH; so much so that no discussion of the hypothesis seems complete today without a well deserved inclusion of some musical reference. In the most recent attempts at a proof of the Riemann Hypothesis, both quantum physics, a study of the energetic scaffold of matter, and chaos theory, also now part of studies of creativity in neurobiology, are playing a role. All this conspires to give RH an “organic” feel.

Given the range and diversity of the disciplines, concepts, and media involved in our attempts to prove the Riemann Hypothesis, all having strong ties with the primes, could RH be hinting at some version of the primes inside the human brain? Andrei Khrennikov's working mathematical models of "the process of thinking," described from the point of view of an interplay between of the conscious mind and the subconscious, in terms of dynamical systems over p-adic fields [4.], may suggest a mathematician's resounding answer to the question in the affirmative. However, move slowly to a conclusion; it could be all too easy to confuse the mimicry of a phenomenon with its actual process. In the context of the Riemann Hypothesis, the subtle path to insight with regards to exceptional creativity likely resides not in forcing a connection between mathematics and the brain, but in beholding the brain's inner workings that are suggested by the mathematics and, conversely, in our appreciating intrinsic, brain-like workings within the mathematics and how they contribute in more or less subtle ways to the beauty we perceive in RH [5.].

Likewise, a detailed examination of works of art can give insights into the employed methods as well as the artists’s mental processes. One such an examination is that of Georges Braque’s paintings, employing x-ray and other technical means, which will be presented in a soon to be on view exhibit by the Mildred Lane Kemper Art Museum at WUSTL. Georges Braque’s experimental still lifes in the 1930s and 1940s (following the neoclassical retour à l’ordre—return to order) are described as bold, large-scale, tactile Cubist spaces that reveal Braque’s manipulation of pigments and materials and his practice of continually reworking canvases, as well as they reveal the painter’s focused attention on still life compositions and the methods and materiality of painting.  For more information on the exhibit titled Georges Braque and the Cubist Still Life, 1928-1945, which will be on view from January 25 through April 21, 2013 at the Kemper, visit http://kemperartmuseum.wustl.edu/exhibitions/Braque»

— Story by Marie C. Taris, http://www.math.wustl.edu/marietaris/math.html»
Posted 1/16/2013

Enjoyed this story? See also December 2012 News story On Undecidability and Millepedes_with Art by A.T. Fomenko»

1. I.M. Gelfand, (1941) Normierte Ringe, Mat. Sbornik, N.S. 9(51), 3-24
1a. I.M. Gelfand, (1941) Über absolut konvergente trigonometrische Reihen und Integrale, Mat.Sbornik, N.S. 9(51), 51–66
2. Marilyn Kochman, (2003) IN PERSON; An Equation for Success, New York Times
3. Don Zagier, (1990) A one-sentence proof that every prime pΞ1(mod 4) is a sum of two squares, American Mathematical Monthly, Volume 97(2), 144
4. A.Yu. Khrennikov A.Yu., (1998) Human subconscious as the p-adic dynamical system, J. of Theor. Biology 193, 179-196.
5. M. Taris, (January 11, 2013) Upon a Lady's Unbinding, preprint.