Using Mathematics to Explain a Scientific Theory

Friend, Michèle; Molinini, Daniele

doi:10.1093/philmat/nkv022

Cited by 7 publications

(3 citation statements)

References 22 publications

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“…By showing the connection between two radically different approaches, Andréka & Németi (2014) takes an important first step in bringing these sporadic axiom systems together. See Friend (2015) and Friend & Molinini (2015) for general discussions of Andréka-Németi school's project and methodology from points of view of epistemological significance and role in scientific explanation.…”

Section: Contributions In This Papermentioning

confidence: 99%

Groups of Worldview Transformations Implied by Einstein’s Special Principle of Relativity Over Arbitrary Ordered Fields

Madarász¹,

Stannett²,

Székely³

2021

The Review of Symbolic Logic

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Section: Contributions In This Papermentioning

confidence: 99%

Groups of Worldview Transformations Implied by Einstein’s Special Principle of Relativity Over Arbitrary Ordered Fields

Madarász¹,

Stannett²,

Székely³

2021

The Review of Symbolic Logic

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“…For discussion of the methodological and epistemological significance of this project, see Friend and Molinini [13,14]. For a comprehensive comparison between Hilbert's project on the foundations of physics and the Andréka-Németi project, see Formica and Friend [12].…”

Section: Introductionmentioning

confidence: 99%

Investigations of isotropy and homogeneity of spacetime in first-order logic

Madarász

Stannett

Székely

2022

Annals of Pure and Applied Logic

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“…The investigation presented in this paper is part of the Andréka-Németi school's general project of logic-based axiomatic foundations of relativity theories, see e.g., [AMN06, AMN07, AMNS12, AN14]. Friend and Molinini [Fri15,FM15] discuss the significance of this project and the underlying methodology from the viewpoints of epistemology and explanation in science. One important feature of using a firstorder logic-based axiomatic framework is that it helps avoid hidden assumptions, which is fundamental in foundational analyses of this nature.…”

Section: Introductionmentioning

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.

show abstract

Using Mathematics to Explain a Scientific Theory

Cited by 7 publications

References 22 publications

Groups of Worldview Transformations Implied by Einstein’s Special Principle of Relativity Over Arbitrary Ordered Fields

Groups of Worldview Transformations Implied by Einstein’s Special Principle of Relativity Over Arbitrary Ordered Fields

Investigations of isotropy and homogeneity of spacetime in first-order logic

Groups of Worldview Transformations Implied by Isotropy of Space

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