Segre and the Foundations of Geometry: From Complex Projective Geometry to Dual Numbers

Marvel et al. [2009] studied sinusoidally coupled phase oscillators, generalizing coupled Josephson junctions. They obtained an explicit reduction of the dynamics to a parametrised family of ODEs on the three-dimensional Möbius group. This differs from the usual reduction on to the orbit space of a symmetry group. We apply the viewpoint of complex projective geometry to obtain an alternative proof that trajectories lie on orbits of the Möbius group, and derive a different explicit form for the reduced ODE. The main innovation is the use of homogeneous coordinates, which linearize the action of the Möbius group and lead to a simple coordinate system in which to write the reduced ODE. We also discuss a Lie-theoretic interpretation.

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Complex Projective Geometry

10.1017/cbo9780511662652 ◽

1992 ◽

Keyword(s):

Projective Geometry ◽

Complex Projective

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HILBERT, DUALITY, AND THE GEOMETRICAL ROOTS OF MODEL THEORY

The Review of Symbolic Logic ◽

10.1017/s1755020317000260 ◽

2017 ◽

Vol 11 (1) ◽

pp. 48-86 ◽

Cited By ~ 2

Author(s):

GÜNTHER EDER ◽

GEORG SCHIEMER

Keyword(s):

Nineteenth Century ◽

19Th Century ◽

Projective Geometry ◽

Model Theory ◽

Euclidean Geometry ◽

Point Of View ◽

Logical Theory ◽

Foundations Of Geometry ◽

Theoretic Point

AbstractThe article investigates one of the key contributions to modern structural mathematics, namely Hilbert’sFoundations of Geometry(1899) and its mathematical roots in nineteenth-century projective geometry. A central innovation of Hilbert’s book was to provide semantically minded independence proofs for various fragments of Euclidean geometry, thereby contributing to the development of the model-theoretic point of view in logical theory. Though it is generally acknowledged that the development of model theory is intimately bound up with innovations in 19th century geometry (in particular, the development of non-Euclidean geometries), so far, little has been said about how exactly model-theoretic concepts grew out of methodological investigations within projective geometry. This article is supposed to fill this lacuna and investigates this geometrical prehistory of modern model theory, eventually leading up to Hilbert’sFoundations.

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