Gauss used to worship the goddess Nature. An intelligent man. And mathematics has never truly bloomed but for the study of nature. Physics and mathematics have been like entwined lovers since Newton, sometimes peaceful and tranquil and at others in friendly conflict, like when Dirac decided to thumb his nose at mathematicians with his adoration of the cult of physics and gave physicists of the rest of the twentieth century bad habits in integrating over path spaces formally and other still bad habits while introducing the function that is zero everywhere and is not quite defined at a point but does integrate to the point. In Dirac’s case, this was excusable. The man was a tremendous genius, as anyone who has looked at any of his papers or his book on Quantum Mechanics will be able to verify. In his book, like all the greatest of geniuses like Ludwig Wittgenstein’s Tractatus Philosophico-Logicus, he has no references. They wrote a bit like how Kafka had declared writing should be done with his tremendous metanoia and childbirth experience with “The Judgment”, which he wrote in a single sitting of eight hours. It was Dirac’s delta function that was formalized successfully primarily by Sergei Sobolev, theory of linear functionals of smooth functions which gave form to the notion of mathematical weak solutions. This brings us to the topic of our note. When Jean Leray in 1933 for the first time proved the existence of weak solutions to the fluid dynamics equations of Navier and Stokes, this was before Sergei Sobolev and Laurent Schwartz had even worked out the theory of distributions. In Leray’s 1933 paper he explicitly mentions that he thought that the Navier-Stokes equation could develop singularities in finite time. So this distribution concept allowed a flexibility for fairly unintuitive behavior if intuition is that from classical theory of functions. Dirac’s boldness, if it be given credit for the naturality of distributions, if it is seen through the lens of a sort of imagination of disorder, captured some part of natural disorder for mathematicians, and I do not have the experience with the specialized mathematics that have explored the depths of distributions. I have much more background with functions that arise with wavelet analysis which we can learn is tightly bound to function classes with heterogeneous behavior, but they are ordinary functions. Nature does produce roughness as it is known from two-dimensional images that there is a Besov scale to nature. Benoit Mandelbrot was responsible to produce a challenge to the use of Markov processes like Brownian motion whose typical paths are nowhere differentiable to fractional Brownian motion in 1968. Donoho and Tanner’s 2005 breakthrough in statistical analysis was a type of disorder that is congestion but not explicitly but rather through telling us that nearly black objects subjected to Gaussian random matrix transformation can only be recovered exactly by convex relaxation until a critical threshold is reached after which the congestion is too much and signal recovery will fail.

The deviation from smooth and nice to various sorts of disorder has been a theme for the entire twentieth century in mathematics, the most impressive campaign that I experienced myself was probably on chaos theory when I was still in high school in the late 1980s. James Gleick’s Chaos: The Making of New Science had material from the stories of Feigenbaum and other physicists and also Mandelbrot’s challenge to Riemannian paradigm of geometry. Of course this is not just Mandelbrot but also Gromov and others who had been studying rougher things.

Nassim Taleb made a splash not just in finance but also in public consciousness with his challenges to the randomness, disorder, and choatic metrics that have lulled us into accepting the status quo regarding the claims of the slaves to money of the science they want to claim in grey conservative accounting suits that the risks are much lower than they actually are because if we go by Gaussian randomness, the possible wildness of not just markets but of many other phenomena are highly underestimated. Gauss did not intend Gaussian random variables to model much besides white noise in scientific measurements. Turbulent hydrodynamics had beenmost hallow sanctuary of the science establishment’s for the fear of the truly chaotic and unpredictable. I will admit that even though I was a good Princeton mathematics student, I was never fully at ease with the new ideas of chaos. They seemed to be a marketing bonanza for a public hungry for cool pictures mostly. This is clearly because I was young, ignorant and myopic. These explorers into the depth of nature, these great geniuses, like Leray looking at singularities, or Cafarelli-Kohn-Nirenberg who proved that the singular set of Navier-Stokes equation with reasonable constraints have Hausdorff dimension at most 1, these people are in technical fields using various notions of smoothness which are generally unknown to the public. In this what is fascinating to me is to consider which notions of smoothness are worthwhile in opening up more facets of nature for us. Of course it is very hard to have public impact: the non-euclidean revolution of the nineteenth century from Kant telling us that Euclid belongs in the human head never did change the public ideas or the school education from Euclid.

on June 1, 2015 at 6:10 pm |ReaderG., please do consider my offer.

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