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@article{BSL_2024_53_1_a2, author = {Gruszczy\'nski, Rafa{\l}}, title = {Mathematical {Methods} in {Region-Based} {Theories} of {Space:} {The} {Case} of {Whitehead} {Points}}, journal = {Bulletin of the Section of Logic}, pages = {63--104}, publisher = {mathdoc}, volume = {53}, number = {1}, year = {2024}, language = {en}, url = {http://geodesic.mathdoc.fr/item/BSL_2024_53_1_a2/} }
TY - JOUR AU - Gruszczyński, Rafał TI - Mathematical Methods in Region-Based Theories of Space: The Case of Whitehead Points JO - Bulletin of the Section of Logic PY - 2024 SP - 63 EP - 104 VL - 53 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/BSL_2024_53_1_a2/ LA - en ID - BSL_2024_53_1_a2 ER -
Gruszczyński, Rafał. Mathematical Methods in Region-Based Theories of Space: The Case of Whitehead Points. Bulletin of the Section of Logic, Tome 53 (2024) no. 1, pp. 63-104. http://geodesic.mathdoc.fr/item/BSL_2024_53_1_a2/
[1] B. Bennett, I. Düntsch, Axioms, algebras and topology, [in:] M. Aiello, I. Pratt-Hartmann, V. Benthem (eds.), Handbook of Spatial Logics, chap. 3, Springer, Dordrecht (2007), pp. 99–159 | DOI
[2] L. Biacino, G. Gerla, Connection structures, Notre Dame Journal of Formal Logic, vol. 32(2) (1991), pp. 242–247 | DOI
[3] L. Biacino, G. Gerla, Connection structures: Grzegorczyk’s and Whitehead’s definitions of point, Notre Dame Journal of Formal Logic, vol. 37(3) (1996), pp. 431–439 | DOI
[4] B. L. Clarke, A calculus of individuals based on ‘connection’, Notre Dame Journal of Formal Logic, vol. 22(3) (1981), pp. 204–217 | DOI
[5] B. L. Clarke, Individuals and points, Notre Dame Journal of Formal Logic, vol. 26(1) (1985), pp. 61–67 | DOI
[6] S. W. Davis, Spaces with linearly ordered local bases, Topology Proceedings, vol. 3 (1978), pp. 37–51.
[7] T. de Laguna, Extensive abstraction: A suggestion, The Philosophical Review, vol. 30(2) (1921), pp. 216–218.
[8] H. de Vries, Compact spaces and compactifications, Van Gorcum and Comp. N.V., Amsterdam (1962).
[9] G. Del Piero, A class of fit regions and a universe of shapes for continuum mechanics, Journal of Elasticity, vol. 70 (2003), pp. 175–195 | DOI
[10] G. Del Piero, A new class of fit regions, Note di Matematica, vol. 27(2) (2007), pp. 55–67.
[11] I. Düntsch, W. MacCaull, D. Vakarelov, M. Winter, Distributive contact lattices: Topological representations, The Journal of Logic and Algebraic Programming, vol. 76(1) (2008), pp. 18–34 | DOI
[12] C. Eschenbach, A Mereotopological Definition of ‘Point’, [in:] C. Eschenbach, C. Habel, B. Smith (eds.), Topological Foundations of Cognitive Science, Graduiertenkolleg Kognitionswissenschaft, Hamburg (1994), pp. 63–80 | DOI
[13] A. Galton, The mereotopology of discrete space, [in:] C. Freksa, D. M. Mark (eds.), COSIT ’99: Proceedings of the International Conference on Spatial Information Theory: Cognitive and Computational Foundations of Geographic Information Science, Springer, Berlin, Heidelberg (1999), pp. 251–266 | DOI
[14] A. Galton, Multidimensional mereotopology, [in:] D. Dubois, C. Welty, M.-A. Williams (eds.), Principles of Knowledge Representation and Reasoning: Proceedings of the Ninth International Conference, AAAI Press (2004), pp. 45–54.
[15] R. Gruszczyński, Niestandardowe teorie przestrzeni, Wydawnictwo Naukowe Uniwersytetu Mikołaja Kopernika, Toruń (2016).
[16] R. Gruszczyński, S. J. Martinez, Grzegorczyk and Whitehead points: the story continues (2023), arxiv:2303.08664.
[17] R. Gruszczyński, A. Pietruszczak, A study in Grzegorczyk point-free topology. Part I: Separation and Grzegorczyk structures, Studia Logica, vol. 106 (2018), pp. 1197–1238 | DOI
[18] R. Gruszczyński, A. Pietruszczak, A study in Grzegorczyk point-free topology. Part II: Spaces of points, Studia Logica, vol. 107 (2019), pp. 809–843 | DOI
[19] R. Gruszczyński, A. Pietruszczak, Grzegorczyk points and filters in Boolean contact algebras, The Review of Symbolic Logic, vol. 16(2) (2023), pp. 509–528 | DOI
[20] A. Grzegorczyk, Axiomatizability of geometry without points, Synthese, vol. 12(2–3) (1960), pp. 228–235 | DOI
[21] T. Hahmann, CODI: A Multidimensional Theory of Mereotopology with Closure Operations, Applied Ontology, vol. 15(3) (2020), pp. 1–61 | DOI
[22] E. Huntington, A set of postulates for abstract geometry, expressed in terms of the simple relation of inclusion, Mathematische Annale, vol. 73 (1913), pp. 522–559 | DOI
[23] T. Ivanova, D. Vakarelov, Distributive mereotopology: extended distributive contact lattices, Annals of Mathematics and Artificial Intelligence, vol. 77 (2016), pp. 3–41 | DOI
[24] P. T. Johnstone, Stone spaces, vol. 3 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (1982).
[25] P. T. Johnstone, The point of pointless topology, Bulletin (New Series) of the American Mathematical Society, vol. 8(1) (1983), pp. 41–53, URL: https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-8/issue-1/The-point-of-pointless-topology/bams/1183550014.full
[26] S. Koppelberg, Handbook of Boolean Algebras, vol. 1, Elsevier, Amsterdam (1989).
[27] T. Lando, D. Scott, A calculus of regions respecting both measure and topology, Journal of Philosophical Logic, vol. 48(5) (2019), pp. 825–850 | DOI
[28] T. Mormann, Continuous lattices and Whiteheadian theory of space, Logic and Logical Philosophy, vol. 6(6) (1998), pp. 35–54 | DOI
[29] J. Picado, A. Pultr, Frames and Locales, Frontiers in Mathematics, Birkhäuser, Basel (2012) | DOI
[30] J. Picado, A. Pultr, Separation in Point-Free Topology, Birkhäuser, Basel (2021) | DOI
[31] A. Pietruszczak, Metamereology, Nicolaus Copernicus University Publishing House, Toruń (2018).
[32] A. Pietruszczak, Foundations of the theory of parthood, Springer, Toruń (2020) | DOI
[33] I. Pratt, O. Lemon, Ontologies for plane, polygonal mereotopology, Notre Dame Journal of Formal Logic, vol. 38(2) (1997), pp. 225–245 | DOI
[34] I. Pratt-Hartmann, Empiricism and rationalism in region-based theories of space, Fundamenta Informaticae, vol. 46(1–2) (2001), pp. 159–186
[35] K. Robering, “The whole is greater than the part.” Mereology in Euclid’s Elements, Logic and Logical Philosophy, vol. 25(3) (2016), pp. 371–409 | DOI
[36] P. Roeper, Region-based topology, Journal of Philosophical Logic, vol. 26(3) (1997), pp. 251–309 | DOI
[37] H. Rüping, An example of a regular but not linear-based topological space (2016), URL: https://mathoverflow.net/q/240345 mathOverflow, version: 2016-06-03.
[38] B. Russell, Our knowledge of the external world, George Allen and Unwin LTD, London (1914).
[39] D. J. Schoop, Points in point-free mereotopology, Fundamenta Informaticae, vol. 46(1–2) (2001), pp. 129–143.
[40] E. V. Shchepin, Real functions and near-normal spaces, Fundamenta Informaticae, vol. 13 (1972), pp. 820–830 | DOI
[41] J. Stell, Boolean connection algebras: A new approach to the Region-Connection Calculus, Artificial Intelligence, vol. 122(1) (2000), pp. 111–136 | DOI
[42] M. H. Stone, The theory of representations for Boolean algebras, Transactions of the American Mathematical Society, vol. 40(1) (1936), pp. 37–111 | DOI
[43] T. Tao, 245b notes 4: The Stone and Loomis-Sikorski representation theorems (optional) (2009), URL: https://terrytao.wordpress.com/2009/01/12/245b-notes-1-the-stone-and-loomis-sikorski-representation-theorems-optional/ accessed: August 01, 2010
[44] A. C. Varzi, Mereology, [in:] E. N. Zalta (ed.), The Stanford Encyclopedia of Philosophy (2016), URL: https://plato.stanford.edu/entries/mereology/ spring 2016 edition.
[45] A. C. Varzi, Points as higher-order constructs, [in:] G. Hellman, S. Shapiro (eds.), The History of Continua: Philosophical and Mathematical Perspectives, Oxford University Press, Oxford (2020), pp. 347–378 | DOI
[46] A. N. Whitehead, Enquiry Concerning the Principles of Human Knowledge, Cambridge University Press, Cambridge (1919).
[47] A. N. Whitehead, The Concept of Nature, Cambridge University Press, Cambridge (1920).
[48] A. N. Whitehead, Process and Reality, MacMillan, New York (1929).
[49] M. Winter, T. Hahmann, M. Grüninger, On the algebra of regular sets, Annals of Mathematics and Artificial Intelligence, vol. 65 (2012), pp. 25–60 | DOI