Abstract
Since the early days of physics, space has called for means to represent, experiment, and reason about it. Apart from physicists, the concept of space has intrigued also philosophers, mathematicians and, more recently, computer scientists. This longstanding interest has left us with a plethora of mathematical tools developed to represent and work with space. Here we take a special look at this evolution by considering the perspective of Logic. From the initial axiomatic efforts of Euclid, we revisit the major milestones in the logical representation of space and investigate current trends. In doing so, we do not only consider classical logic, but we indulge ourselves with modal logics. These present themselves naturally by providing simple axiomatizations of different geometries, topologies, space-time causality, and vector spaces.
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Acknowledgements
We thank Elisabetta Pallante for fruitful discussion on the theory of quantum gravity and de Sitter space-time. We are also grateful to the referees for the suggestions which improved the presentation of the paper.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Aiello, M., Bezhanishvili, G., Bloch, I. et al. Logic for physical space. Synthese 186, 619–632 (2012). https://doi.org/10.1007/s11229-011-9913-9
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DOI: https://doi.org/10.1007/s11229-011-9913-9