Abstract
Formal geometry is a fundamental tool for showing how relevant metric qualities, such as depths, lengths, and volumes, as well as location concepts, such as points, can be constructed from experience. The ontological challenge of information grounding lies in the choice of concepts to consider as primitive, vs. those to be constructed. It also lies in accounting for the relativity and finiteness of experiential space. The grounding approach proposed here constructs geometrical concepts from primitives of the human attentional apparatus for guiding attention and performing perceptual operations. This apparatus enables humans to take attentional steps in their perceived vista environment and to perform geometric comparisons. We account for the relativity of experienced space by constructing locations relative to a reference frame of perceived point-like features. The paper discusses perceptual operations and the idea of point-like features, and introduces a constructive calculus that reflects the generation of domains of geometric comparison from the perspective of an observer. The calculus is then used to construct a model and to motivate an axiomatization of absolute geometry in a finite relativist flavour.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aristotle: Physics. Hardie, R. P., Gaye, R. K.(eds.) University of Adelaide, Adelaide (2007), http://ebooks.adelaide.edu.au/a/aristotle/physics/
Arnheim, R.: Art and visual perception. University of California Press, Berkeley (2004) (50th anniversary printing edn.)
Bennett, B.: A categorical axiomatisation of region-based geometry. Fundamenta Informaticae 46(1-2), 145–158 (2001)
Burgess, N.: Spatial cognition and the brain. Ann. N. Y. Acad. Sci. 1124, 77–97 (2008)
Dummett, M.: The logical basis of metaphysics. Duckworth, London (1991)
Eddington, A.: What is geometry? In: Smart, J. (ed.) Problems of Space and Time, pp. 163–177. The Macmillan Company, New York (1964)
Gerla, G., Volpe, R.: Geometry without points. The American Mathematical Monthly 92(10), 707–711 (1985)
Gibson, J.: The ecological approach to visual perception. Houghton Mifflin, Boston (1979)
van Glasersfeld, E.: An attentional model for the conceptual construction of units and number. Journal for Research in Mathematics Education 12(2), 83–94 (1981)
van Glasersfeld, E.: Radical Constructivism: A Way of Knowing and Learning. The Falmer Press, London (1995)
Habel, C.: Discreteness, finiteness, and the structure of topological spaces. In: Topological foundations of cognitive science. Papers from the workshop at the FISI-CS, Buffalo, NY. Graduiertenkolleg Kognitionswissenschaft (Report 37), pp. 81–90. Universität Hamburg, Hamburg (1994)
Harnad, S.: The symbol grounding problem. Physica D 42, 335–346 (1990)
Hilbert, D.: Grundlagen der Geometrie, 12th edn. (1977)
Janowicz, K., Compton, M.: The stimulus-sensor-observation ontology design pattern and its integration into the semantic sensor network ontology. In: Proceedings of the 3rd International Workshop on Semantic Sensor Networks, SSN 2010 (2010)
Köhler, W.: Gestalt psychology. An introduction to new concepts in modern psychology. Liveright, New York (1992)
Kuhn, W.: Semantic reference systems. Int. J. Geogr. Inf. Science 17(5), 405–409 (2003)
Langacker, R.: Nouns and verbs. Language 63(1), 53–94 (1987)
Leibniz, G., Clarke, S.: The Leibniz-Clarke correspondence. Manchester University Press, Manchester (1956)
Lorenzen, P.: Einführung in die operative Logik und Mathematik. Springer, Berlin (1955)
Lorenzen, P.: Das aktual-unendliche in der mathematik. Philosophia Naturalis 4(1), 3–11 (1957)
Malkevitch, J.: Finite geometries? (2006), http://www.ams.org/featurecolumn/archive/finitegeometries.html
Marchetti, G.: A presentation of attentional semantics. Cognitive Processsing 7(3), 163–194 (2006)
Mark, D., Frank, A.: Experiential and formal models of geographic space. Environment and Planning B 23, 3–24 (1996)
Masolo, C., Borgo, S., Gangemi, A., Guarino, N., Oltramari, A.: Wonderweb deliverable d18: Ontology library, Trento, Italy (2003)
Montello, D.R.: Scale and multiple psychologies of space. In: Campari, I., Frank, A.U. (eds.) COSIT 1993. LNCS, vol. 716, pp. 312–321. Springer, Heidelberg (1993)
Newell, A., Simon, H.: Computer science as empirical inquiry: Symbols and search. Commun. ACM 19(3), 113–126 (1976)
Piaget, J.: Genetic epistemology. Woodbridge lecture no. 8, 1st edn. Columbia University Press, New York (1970)
Poincaré, H.: Science and hypothesis. Dover Publ., N.Y (1952)
Prawitz, D.: Ideas and results in proof theory. In: Fenstad, J. (ed.) Proc. 2nd Scandinavian Logic Symposium, pp. 237–309. North-Holland, Amsterdam (1971)
Probst, F., Espeter, M.: Spatial dimensionality as classification criterion for qualities. In: Bennett, B., Fellbaum, C. (eds.) Formal Ontology in Information Systems: Proceedings of the Fourth International Conference: FOIS 2006. Frontiers in Artificial Intelligence and Applications, vol. 150, pp. 77–88. IOS Press, Amsterdam (2006)
Pylyshyn, Z.: Things and places. How the mind connects with the world. The MIT Press, Cambridge (2007)
Quine, W.: Two dogmas of empiricism. The Philosophical Review 60, 20–43 (1951)
Quine, W.: The roots of reference. Open Court Publishing, La Salle (1974)
Quine, W.: On what there is. In: From a Logical Point of View. 9 Logico-Philosophical Essays, 2nd edn. Harvard University Press, Cambridge (1980)
Rizzolatti, G., Sinigaglia, C.: Mirrors in the brain: How our minds share actions and emotions. Oxford University Press, Oxford (2008)
Scheider, S.: Grounding geographic information in perceptual operations. Ph.D. thesis, University of Münster (2011)
Scheider, S., Janowicz, K., Kuhn, W.: Grounding geographic categories in the meaningful environment. In: Hornsby, K., Claramunt, C., Denis, M., Ligozat, G. (eds.) COSIT 2009. LNCS, vol. 5756, pp. 69–87. Springer, Heidelberg (2009)
Scheider, S., Probst, F., Janowicz, K.: Constructing bodies and their qualities from observations. In: Proc. of the Sixth International Conference on Formal Ontology in Information Systems (FOIS 2010), pp. 131–144. IOS Press, Amsterdam (2010)
Scholl, B.: Objects and attention: The state of the art. Cognition 80, 1–46 (2001)
Schroeder-Heister, P.: Lorenzen’s operative justification of intuitionistic logic. In: van Atten, M., Boldini, P., Bourdeau, M., Heinzmann, G. (eds.) One Hundred Years of Intuitionism (1907-2007), Birkhäuser, Basel (2008)
Schwabhäuser, W., Szmielew, W., Tarski, A.: Metamathematische Methoden in der Geometrie, Teil I: Ein axiomatischer Aufbau der euklidischen Geometrie. Springer, Berlin (1983)
Suppes, P.: Finitism in geometry. Erkenntnis 54, 133–144 (2001)
Tarski, A.: Foundations of the geometry of solids. In: Tarski, A., Woodger, J. (eds.) Logic, Semantics, Metamathematics: Papers from 1923 to 1938, pp. 24–30. Clarendon Press, Oxford (1956)
Tarski, A.: What is elementary geometry. In: Henkin, P.S.L., Tarski, A. (eds.) The Axiomatic Method. With Special Reference to Geometry and Physics, pp. 16–29. North-Holland Publishing, Amsterdam (1959)
Tarski, A., Givant, S.: Tarski’s system of geometry. The Bulletin of Symbolic Logic 5(2), 175–214 (1999)
Tomasello, M.: The cultural origins of human cognition. Harvard University Press, Cambridge (1999)
Tversky, B.: Structures of mental spaces: How people think about space. Environment and Behaviour 35(1), 66–80 (2003)
Vaihinger, H.: Die Philosophie des Als Ob. System der theoretischen, praktischen und religiösen Fiktionen der Menschheit auf Grund eines idealistischen Positivismus. VDM Verlag Dr. Müller, Saarbrücken (2007)
VanRullen, R., Koch, C.: Is perception discrete or continuous? Trends in Cognitive Sciences 7(5), 207–213 (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Scheider, S., Kuhn, W. (2011). Finite Relativist Geometry Grounded in Perceptual Operations. In: Egenhofer, M., Giudice, N., Moratz, R., Worboys, M. (eds) Spatial Information Theory. COSIT 2011. Lecture Notes in Computer Science, vol 6899. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23196-4_17
Download citation
DOI: https://doi.org/10.1007/978-3-642-23196-4_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-23195-7
Online ISBN: 978-3-642-23196-4
eBook Packages: Computer ScienceComputer Science (R0)