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Poznanie geometryczne z kognitywnego punktu widzenia

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Pogonowski J.

Zagadnienia Filozoficzne w Nauce

2021

issue 70

183-211

This review discusses the content of Mateusz Hohol’s new book Foundations of Geometric Cognition. Mathematical cognition has until now focused mainly on human numerical abilities. Hohol’s work tackles geometric cognition, an issue that has not been described in previous investigations into mathematical cognition. The main strength of the book lies in its critical analysis of a huge amount of results from empirical experiments. The author formulates his theoretical proposals very carefully, avoiding radical and one-sided solutions. He claims that human geometric cognition is based mainly on two core systems, both being phylogenetically hardwired, namely the system of layout geometry and the system of object geometry. The interaction of these systems becomes amplified in the individual development of the mind, which, in turn, is supported by the use of language. The second part of the review contains the reviewer’s remarks concerning the history of geometry, experiments related to spatial representations, and the role of geometry in mathematical education.

Regions-based two dimensional continua: The Euclidean case

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Hellman G., Shapiro S.

Logic and Logical Philosophy

2015

vol. 24

issue 4 Mereology and Beyond

499-534

We extend the work presented in [7, 8] to a regions-based, two-dimensional, Euclidean theory. The goal is to recover the classical continuum on a point-free basis. We first derive the Archimedean property for a class of readily postulated orientations of certain special regions, “generalized quadrilaterals” (intended as parallelograms), by which we cover the entire space. Then we generalize this to arbitrary orientations, and then establishing an isomorphism between the space and the usual point-based R × R. As in the one-dimensional case, this is done on the basis of axioms which contain no explicit “extremal clause” (to the effect that “these are the only ways of generating regions”), and we have no axiom of induction other than ordinary numerical (mathematical) induction. Finally, having explicitly defined ‘point’ and ‘line’, we will derive the characteristic Parallel’s Postulate (Playfair axiom) from regions-based axioms, and point the way toward deriving key Euclidean metrical properties.

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1 Logic and Logical Philosophy

1 Zagadnienia Filozoficzne w Nauce

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1 Pogonowski J.

1 Shapiro S.

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Poznanie geometryczne z kognitywnego punktu widzenia

Regions-based two dimensional continua: The Euclidean case