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Summary: A Modal Walk Through Space
Marco Aiello and Johan van Benthem
Institute for Logic, Language and Computation
University of Amsterdam
Plantage Muidergracht 24, 1018 TV Amsterdam
The Netherlands
E-mail: faiellom,johang@science.uva.nl
Web: http://www.science.uva.nl/~{aiellom,johan}
Abstract
We investigate the major mathematical theories of space from a modal
standpoint: topology, aĆne geometry, metric geometry, and vector alge-
bra. This allows us to see new ne-structure in spatial patterns which
suggests analogies across these mathematical theories in terms of modal,
temporal, and conditional logics. Throughout the modal walk through
space, expressive power is analyzed in terms of language design, bisim-
ulations, and correspondence phenomena. The result is both uni cation
across the areas visited, and the uncovering of interesting new questions.
1 Logics of Space
The study of Space has been a driving force in the foundations of mathemat-
ics, witness the history of formal proof, axiomatic theories, or the traditional
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