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.

Hermann Graßmann: Un poli-matemático extraodinario

En este artículo analizamos algunos aspectos de la vida de Hermann Grassmann como poli-matemático. Citamos como un ejemplo interesante su teoría de mezcla de los colores.

The Emergence of Foundational Study

This chapter looks at how modern foundational study has twofold mathematical roots. One is the discovery of non-Euclidean geometries, especially the proof of independence of the parallel postulate by Eugenio Beltrami in 1868, in his Saggio di interpretazione della geometria non-euclidea (Treatise on the interpretation of non-Euclidean geometry). The other root is arithmetical, retraceable through Peano and others to the 1861 book Lehrbuch der Arithmetik für höhere Lehranstalten (Arithmetic for higher institutions of learning) by the high school teacher Hermann Grassmann. In each of these two cases, one has to set things straight: To prove independence in geometry, one has to ask what the axioms are, and maybe even the principles of proof.

From Religion to Dialectics and Mathematics

Hermann Grassmann is known to be the founder of modern vector and tensor calculus. Having as a theologian no formal education in mathematics at a university he got his basic ideas for this mathematical innovation at least to some extent from listening to Schleiermacher’s lectures on Dialectics and, together with his brother Robert, reading its publication in 1839. The paper shows how the idea of unity and various levels of reality first formulated in Schleiermacher’s talks about religion in 1799 were transformed by him into a philosophical system in his dialectics and then were picked up by Grassmann and operationalized in his philosophical-mathematical treatise on the extension theory (German: Ausdehnungslehre) in 1844.