Bernhard Riemann, the Ear, and an Atom of Consciousness

Why did Bernhard Riemann (1826–1866), arguably the most original mathematician of his generation, spend the last year of life investigating the mechanism of hearing? Fighting tuberculosis and the hostility of eminent scientists such as Hermann Helmholtz, he appeared to forsake mathematics to prosecute a case close to his heart. Only sketchy pages from his last paper remain, but here we assemble some significant clues and triangulate from them to build a broad picture of what he might have been driving at. Our interpretation is that Riemann was a committed idealist and from this philosophical standpoint saw that the scientific enterprise was lame without the “poetry of hypothesis”. He believed that human thought was fundamentally the dynamics of “mind-masses” and that the human mind interpenetrated, and became part of, the microscopic physical domain of the cochlea. Therefore, a full description of hearing must necessarily include the perceptual dimensions of what he saw as a single manifold. The manifold contains all the psychophysical aspects of hearing, including the logarithmic transformations that arise from Fechner’s law, faithfully preserving all the subtle perceptual qualities of sound. For Riemann, hearing was a unitary physical and mental event, and parallels with modern ideas about consciousness and quantum biology are made. A unifying quantum mechanical model for an atom of consciousness—drawing on Riemann’s mind-masses and the similar “psychons” proposed by Eccles—is put forward.

Free Love in Many Dimensions

Higher dimensions were explored by mathematicians Bernhard Riemann and Ludwig Schläfli in the middle years of the nineteenth century. Charles Howard Hinton was the key figure to popularize the notion of higher dimensions, and Chapter 17 introduces the idea of a fourth spatial dimension and the methods devised by Hinton to make higher dimensions accessible. Hinton’s books were very influential in the late Victorian and Edwardian periods, and H. G. Wells’ The Time Machine was inspired by Hinton’s ideas. A simplified version of Hinton’s methods was published in a famous short story, Flatland, written by the Reverend Edwin A. Abbott. Following conviction and imprisonment for bigamy, Hinton left England with his family and eventually settled in America.

8. How to win a million dollars

What is the Riemann hypothesis, and why does it matter? ‘How to win a million dollars’ looks in detail at Riemann’s conjecture. While Gauss attempted to explain why primes thin out, Bernhard Riemann in 1859 proposed an exact formula for the distribution of primes, employing Euler’s ‘zeta function’ and the idea of complex numbers. In 2000, the Clay Mathematics Institute offered a million dollars for the solutions of each of seven famous problems, of which the Riemann hypothesis was one. The Riemann hypothesis implies strong bounds on the growth of other arithmetic functions, in addition to the primes-counting function. It remains one of the most famous unsolved problems of mathematics.

Pierre Duhem: Um Filósofo do Senso Comum

O presente artigo visa a elucidar os fundamentos da metodologia científica de Pierre Duhem, realçando alguns aspectos anti-convencionalistas da mesma. Argumentamos que seu método ampara-se em noções e princípios provenientes do senso comum. Inicialmente, distinguimos os significados que este conceito assume ao longo de sua obra, comparando-o com a noção de bom senso, para, em seguida, justificarmos por que suas críticas a Wilhelm Ostwald, Albert Einstein e Bernhard Riemann, feitas em nome do senso comum, não envolvem, como alguns importantes estudiosos supuseram, contradição alguma. Por fim, sustentamos que sua obra de maturidade, especialmente A ciência alemã, apesar de resultante do clima intelectual belicoso, deve ser alçada ao mesmo patamar de importância geralmente atribuído a A teoria física.

Geometry on my Mind

This chapter reviews the history of modern geometry with a focus on the topics that provided the foundation for the new visualization of physics. It begins with Carl Gauss and Bernhard Riemann, who redefined geometry and identified the importance of curvature for physics. Vector spaces, developed by Hermann Grassmann, Giuseppe Peano and David Hilbert, are examples of the kinds of abstract new spaces that are so important for modern physics, such as Hilbert space for quantum mechanics. Fractal geometry developed by Felix Hausdorff later provided the geometric language needed to solve problems in chaos theory. Motion cannot exist without space—trajectories are the tracks of points, mathematical or physical, through it.