On Meaningful Scientific Laws

Falmagne, Jean‐Claude; Doble, Christopher

doi:10.1007/978-3-662-46098-6

Cited by 10 publications

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“…The examples described here and in several other publications (see e.g. [2][3][4]), suggest that, in a long term, and assuming that many other examples of meaningful laws could be obtained, it might be possible to create a more or less comprehensive catalogue of possible laws. Such a catalogue might be useful to a scientist searching for a mathematical formula describing a phenomenon that is investigated.…”

Section: Toward a Catalogue Of Possible Meaningful Lawsmentioning

confidence: 68%

“…The indices 1,1 signify that its variables are expressed in the initial units. 3 The domain and the range of any code F α,β may be affected by the units.…”

Section: Definitionmentioning

confidence: 99%

“…(We expand on this in the last section of this paper. For general introductions and other examples, see [2,3]).…”

mentioning

confidence: 99%

“…Theorem 13 is related to Theorem 7.2.1 in [3], arising from obtaining Eq. 7by assuming that a single code is permutable with respect to the initial code.…”

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confidence: 99%

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On meaningful transformation equations

Falmagne

Doble²

2019

Aequat. Math.

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The meaningfulness condition, applied to scientific or geometric laws, requires that the mathematical form of an equation does not change when we change the units of its ratio scale variables. Suitably formalized, this condition considerably limits the possible form of a law. In this paper, we give five new examples of such restricted representations. We use the meaningfulness condition on the five transformation equations displayed in the left column of the table below, in which x, y, z, u and v are real numbers and F , K, G, H, M and N are real valued functions. We show that under relatively weak general conditions (such as continuity, symmetry, monotonicity, homogeneity), each transformation equation must have, up to some real constants, the representation in the right column. G(x,y)=(x θ +y θ) 1 θ F (G(x,y),z)=H(x,K(y,z)) F (G(x,y),z) =φxyz γ =H(x,K(y,z)) F (G(x,y),H(u,v))=K(M (x,u),N (y,v)) F (G(x,y),H(u,v)) = (x θ +y θ +u θ +v θ) 1 θ = K(M (x,u),N (y,v)) Mathematics Subject Classification. 03A10. Meaningfulness is a fundamental condition for scientific or geometric laws expressed in terms of ratio scale variables: it formally requires that the mathematical form of an equation does not change when we change the units of 814

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Section: Toward a Catalogue Of Possible Meaningful Lawsmentioning

confidence: 68%

“…The indices 1,1 signify that its variables are expressed in the initial units. 3 The domain and the range of any code F α,β may be affected by the units.…”

Section: Definitionmentioning

confidence: 99%

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On meaningful transformation equations

Falmagne

Doble²

2019

Aequat. Math.

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“…Perhaps it was this focus on measurement in the study of one-dimensional sensory stimuli that made it hard for symmetry to spread into the rest of Cognitive Psychology. Note that symmetry and invariance have continued to play an important role in Mathematical Psychology as evidenced by the work of Narens (2007) and Falmagne (Falmagne and & Doble, 2015) on meaningfulness of scientific laws, and more recently, by the work of Vigo (2015) on categorization.…”

Section: Symmetry In Physics and Its Implications Symmetry In Nature mentioning

confidence: 99%

Unifying Physics and Psychophysics on the Basis of Symmetry, Least-Action ≈ Simplicity Principle, and Conservation Laws ≈ Veridicality

Pizlo

2019

The American Journal of Psychology

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Psychophysics refers to the branch of Experimental Psychology that deals with the study of Sensation and Perception. A consensus has grown up among experts in Psychophysics during the last hundred years that the human being's percepts are inferences that are based on a minimum, or simplicity principle, which is applied to the currently available sensory data. These educated guesses play the critical role in establishing veridical perceptual representations of the 3D environment, where by "veridical" we mean that the percept agrees with what is "out there." These veridical representations cannot be achieved without making use of symmetries, much like those known in Physics, where they are essential for characterizing our physical world and deriving the conservation laws. But, unlike in Physics, the important role that symmetry plays in Psychophysics has only been demonstrated and explained within the last ten years. Symmetries represent regularities in our physical world. These symmetries also serve as the source of the redundancies that are inherent in 3D objects and that make vision possible. The main goal of this paper is to show that the similarity between the mathematical formalisms used in Physics and in Psychophysics is not coincidental, and that exploring this similarity can benefit the sciences called Perception and Cognition. This paper includes a brief tutorial about symmetry groups and their relationship to transformation groups as well as to their invariants. It was included to make this material available to readers who are not familiar with these topics.

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Universal Law

Shaw¹

2018

Three Worlds of Collective Human Experience: Individual Life, Social Change, and Human Evolution

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On Meaningful Scientific Laws

Cited by 10 publications

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On meaningful transformation equations

On meaningful transformation equations

Unifying Physics and Psychophysics on the Basis of Symmetry, Least-Action ≈ Simplicity Principle, and Conservation Laws ≈ Veridicality

Universal Law

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