Metric Geometries in an Axiomatic Perspective

Pambuccian, Victor; Struve, Horst; Struve, Rolf

doi:10.1007/978-3-319-60039-0_14

Cited by 10 publications

(2 citation statements)

References 73 publications

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“…Despite the fact that no direct influence has been shown so far, Pieri's approach has proved to be extraordinarily fruitful. In particular, Johannes Hjelmslev's approach, continued by Arnold Schmidt, and finally by Friedrich Bachmann, created an axiomatization of absolute geometry based entirely on "motions", see [Pambuccian, Struve et al 2017], [Marchisotto, Rodríguez-Consuegra et al 2020, § 7.5].…”

Section: Pieri's Strategies For Teaching Demonstrative Geometry: Hypothetical-deductive Systems and Transformationsmentioning

confidence: 99%

Untitled

2021

philosophiascientiae

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Dans les études historiques contemporaines, les contributions de Peano sont généralement envisagées dans le cadre de la tradition logiciste initiée par Frege. Dans cet article, je vais d'abord démontrer que Frege et Peano ont développé de manière indépendante des approches semblables visant à s'appuyer sur la logique pour exprimer rigoureusement des lois mathématiques et les prouver. Ensuite, je soutiendrai cependant que Peano a également utilisé sa logique mathématique d'une manière qui anticipait la formalisation des théories mathématiques, laquelle est incompatible avec la conception de la logique défendue par Frege.

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Section: Pieri's Strategies For Teaching Demonstrative Geometry: Hypothetical-deductive Systems and Transformationsmentioning

confidence: 99%

Untitled

2021

philosophiascientiae

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“…Observers are interpreted to be labels for inertial coordinate systems. Quantities are used to specify coordinates, lengths and related quantities, and we assume that 1 Metric geometries corresponding to these two structures also appear among Cayley-Klein geometries; see, e.g., [Str16] and [PSS17,§6].…”

Section: Frameworkmentioning

confidence: 99%

Groups of Worldview Transformations Implied by Isotropy of Space

Madarász¹,

Stannett²,

Székely³

2020

Preprint

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Given any Euclidean ordered field, Q, and any 'reasonable' group, G, of (1+3)-dimensional spacetime symmetries, we show how to construct a model M G of kinematics for which the set W of worldview transformations between inertial observers satisfies W = G. This holds in particular for all relevant subgroups of Gal cPoi, and cEucl (the groups of Galilean, Poincaré and Euclidean transformations, respectively, where c ∈ Q is a model-specific parameter corresponding to the speed of light in the case of Poincaré transformations).In doing so, by an elementary geometrical proof, we demonstrate our main contribution: spatial isotropy is enough to entail that the set W of worldview transformations satisfies either W ⊆ Gal, W ⊆ cPoi, or W ⊆ cEucl for some c > 0. So assuming spatial isotropy is enough to prove that there are only 3 possible cases: either the world is classical (the worldview transformations between inertial observers are Galilean transformations); the world is relativistic (the worldview transformations are Poincaré transformations); or the world is Euclidean (which gives a nonstandard kinematical interpretation to Euclidean geometry). This result considerably extends previous results in this field, which assume a priori the (strictly stronger) special principle of relativity, while also restricting the choice of Q to the field R of reals.As part of this work, we also prove the rather surprising result that, for any G containing translations and rotations fixing the time-axis t, the requirement that G be a subgroup of one of the groups Gal, cPoi or cEucl is logically equivalent to the somewhat simpler requirement that, for all g ∈ G: g[t] is a line, and if g[t] = t then g is a trivial transformation (i.e. g is a linear transformation that preserves Euclidean length and fixes the time-axis setwise).2010 Mathematics Subject Classification. Primary 51P05 (83A05, 70B99); 20A15; 46B20.7 This can be expressed in our formal language without any assumption about the structure of quantities as: R(4 , here R( p ) 2 , R( p ) 3 , and R( p ) 4 denotes the second, third and fourth component of R( p ) ∈ Q 4 , i.e. if R( p ) = (t, x, y, z), then R( p ) 2 = x, R( p ) 3 = y, and R( p ) 4 = z.

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Comprehending the Connection of Things: Bernhard Riemann and the Architecture of Mathematical Concepts

Plotnitsky

2017

From Riemann to Differential Geometry and Relativity

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Metric Geometries in an Axiomatic Perspective

Cited by 10 publications

References 73 publications

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Groups of Worldview Transformations Implied by Isotropy of Space

Comprehending the Connection of Things: Bernhard Riemann and the Architecture of Mathematical Concepts

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