×

Particle-free bodies and point-free spaces. (English) Zbl 1423.54006

Int. J. Eng. Sci. 72, 155-176 (2013); corrigendum ibid. 79, 81 (2014).
Summary: Few notions in mathematics and physics are as fundamental and useful as the notion of a “point”. However, in addition to the concept of a “point” being far from apparent, the concept is not suitable for describing several important problems in natural philosophy. A far more tangible and sensible idea that is immediately grasped by our mind is that of a “chunk” (a solid object) which seems ideally suited to describe precisely those problems which the notion of “point” seems to hinder. In this paper, I articulate the need for the use of topologies which are “point free” for the proper resolution of many important problems in natural philosophy.

MSC:

54A05 Topological spaces and generalizations (closure spaces, etc.)
00A30 Philosophy of mathematics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bacon, R., Opus majus, (1267), (the opus majus of roger bacon, part 1, page 4, translated by robert belle burke), (1928), University of Pennsylvania Press Philadelphia
[2] Barnes, J. (Ed.). (1995). The complete works of Aristotle (Vol. 1). Princeton: Princeton University Press.
[3] Berkeley, G. (1710). A treatise containing the principles of human knowledge. Printed by Aaron Rhames, for Jeremy Pepyat, Bookseller in Skinner-Row, Dublin (published by Jacob Tonson, London (1734)).
[4] Berkeley, G. (1713). Three dialogues between Hylas and Philonous in opposition to Sceptics and Atheists. In: C. W. Eliot (Ed.), Harvard Classics (Vol. 37, Part 2). New York: P. F. Collier and Sons; 1910.
[5] Birkhoff, G. D., Lattice theory, Vol. 25, (1966), American Mathematical Society Providence
[6] Bobenko, A. I., Discrete differential geometry, (Bobenko, A. I.; Schroder, P.; Sullivan, J. M.; Ziegler, G. M., Oberwolfach seminars, Vol. 38, (2008)), 3-35
[7] Bohm, D., Causality and chance in modern physics, (1957), Routledge & Kegan Paul, D. Van Nostrand Princeton
[8] Bohm, D.; Hilley, B. J., The undivided universe, (1993), Routledge New York · Zbl 0990.81503
[9] Bolzano, B., Paradoxes of the infinite, (1950), Routledge & Keegan Paul London, [trans. by Steele, D. A.]
[10] Boscovich, R. G., A theory of natural philosophy (tran. by J.M. childs), (1966), MIT Press Cambridge
[11] Boussinesq, J., Application des potentiels a L’etude de l’equilibre et due mouvement des solides elastique, (1885), Gauthier Villars Paris · JFM 17.0952.01
[12] Boyer, C., History of calculus, (1959), Dover Publications New York
[13] Bulicek, M., Malek, J., Rajagopal, K. R., & Walton, J. (2012). Existence of solution for anti-plane stress for a new class of strain limiting elastic bodies. Charles University, Prague [preprint MORE]. · Zbl 1329.35302
[14] Burgers, J. M., Mechanical considerations model systems phenomenological theories of relaxation and viscosity, (First report on viscosity and plasticity, (1939), Nordemann Publishing New York), 5-67
[15] Cartwright, R., (Scattered objects, Philosophical essays, (1987), MIT Press Cambridge)
[16] Clarke, B. L., A calculus of individuals based on connection, Notre Dame J. Formal Logic, 22, 204-218, (1982) · Zbl 0438.03032
[17] Clifford, W. K., On the space theory of matter, (Tucker, R., Mathematical papers, (1882), MacMillan and Company London)
[18] Clough, A. H. (Ed.). (1934). Plutarch, lives: Vol. 5. Modern library. New York [translated by John Dryden].
[19] Cohen, P. J., The independence of the continuum hypothesis, Proc. Natl. Acad. Sci. USA, 50, 1143-1148, (1963) · Zbl 0192.04401
[20] Cohen, P. J., The independence of the continuum hypothesis II, Proc. Natl. Acad. Sci. USA, 51, 105-110, (1964)
[21] Collingwood, R. G., The idea of nature, (1986), Greenwood Press Westport
[22] Cosserat, E., & Cosserat, F. (1909). Theorie des corps deformables, Paris, Librairie Scientific A. Hermann et Fils. 953-1173 of Chwolson’s Traite de Physique, 2nd ed., Paris. [Translated in “Theory of deformable bodies, NASA TT F-11, 561, Washington, DC (1968), Clearing house for US Federal Scientific and Technical Information, Springfield, Virginia]. · JFM 40.0862.02
[23] DasGupta, S., A history of Indian philosophy (in five volumes), Delhi, varanasi, patna: motilal banarsidas, (1975), Cambridge University Press, 1922 Cambridge
[24] de Laguna, T., Point, line, and surface as sets of a solid, Journal of Philosophy, 19, 449-461, (1922)
[25] de Laguna, T., Nature of space - part I, Journal of Philosophy, 19, 393-407, (1922)
[26] de Laguna, T., Nature of space - part II, Journal of Philosophy, 19, 421-440, (1922)
[27] Dryden, J., The hind and the panther. A poem in three parts, (1687), Jacob Tonson London
[28] Feller, W. 1950. An introduction to probability theory and its application (vol. 1). New York, London: John Wiley & Sons Inc., Chapman Hall. · Zbl 0039.13201
[29] Galilei, G. (1658). Discorzi e Dimostrazioni matematiche. (Translated by Henry Crew and Alfanso de Salvio), Dialogues concerning the two sciences. New York: Dover Publications, 1954.
[30] Gerla, G.; Volpe, D., Geometry without points, American Mathematical Monthly, 29, 707-711, (1985) · Zbl 0596.51006
[31] Grzegorczyk, A. (1961). Axiomatizability of geometry without points. In Proceedings of the colloquium at Utrecht, January, 1960, Dordrecht (pp. 104-111). · Zbl 0201.32104
[32] Grunbaum, A., A consistent conception of the linear extended continuum as an aggregate of unextended elements, Philosophy of Science, 19, 288-306, (1952) · Zbl 0048.00504
[33] Heath, T. L., A History of Greek Mathematics, Volume II, from Aristarchus to Diophantus, (1921), Oxford University Press London · JFM 48.0046.01
[34] Huntington, E. V., A set of postulates for abstract geometry exposed in terms of the simple relation of inclusion, Mathematische Annalen, lxxiii, 522-529, (1916) · JFM 44.0544.03
[35] Johnstone, P. T., The point of pointless bodies, Bulletin of the American Mathematical Society, 8, 41-53, (1983) · Zbl 0499.54002
[36] Johnstone, P. T., (1991). The art of pointless thinking: a student’s guide to the category of locales. In Proc. Workshop Bremen 1990, Research and Exposition in Math. Category theory at work, Berlin: Helderman Verlag. Vol. 18, pp. 85-107. · Zbl 0745.18003
[37] Johnstone, P. T., Elements of the history of locale theory, (Handbook of the history of general topology, 3, (2001), Kluwer Academic Publishing Dodrecht), 835-851 · Zbl 1001.54001
[38] Johnstone, P. T., Stone spaces, Cambridge studies in advanced mathematics, Vol. 3, (1982), Cambridge University Press Cambridge
[39] Kline, M., Mathematics, the loss of certainty, (1980), Oxford University Press New York
[40] Kulvait, V.; Malek, J.; Rajagopal, K. R., Anti-plane stress for a plate with V-notch for a new class of elastic solids, International Journal of Fracture, 179, 59-73, (2013)
[41] Laplace, S. P. (1951). (Essai Philosophique sur les Probabilités, Mme Ve Courcier, Imprimeur-Libraire pour les Mathématiques, quai des Augustins, Paris (1814)). A philosophical essay on probabilities. Dover Publications. New York [translated into English from the original French 6th ed. by Truscott, F. W. & Emory, F. L.]. · JFM 11.0242.01
[42] MacLane, S.; Birkhoff, G. D., Algebra, (1967), The MacMillan and Company London · Zbl 0153.32401
[43] McKinsey, J. C.C.; Tarski, A., The algebra of topology, Annals of Mathematics, 45, 141-191, (1944) · Zbl 0060.06206
[44] McKinsey, J. C.C.; Tarski, A., On closed elements in closure algebras, Annals of Mathematics, 47, 122-162, (1946) · Zbl 0060.06207
[45] Merriam-Webster Dictionary. (2006). Merriam-Webster Inc., Springfield, MA.
[46] Merton, R. K., On the shoulder of giants, (1993), University of Chicago Press Chicago
[47] Newton, I., Philosophiae naturalis principia Mathematica (1687), (translated by andrew motte), (1995), Prometheus Books New York
[48] Nicod, J. (1924). La geometrie dans le monde sensible, Paris. · JFM 50.0030.03
[49] Noll, W., A mathematical theory of the mechanical behavior of continuous media, Archive for Rational Mechanics and Analysis, 2, 197-226, (1958) · Zbl 0083.39303
[50] Noll, W., A new theory of simple materials, Archive for Rational Mechanics and Analysis, 48, 1-50, (1972) · Zbl 0271.73006
[51] Noll, W., Lectures on the foundations of continuum mechanics and thermodynamics, Archive of Rational Mechanics and Analysis, 52, 82-92, (1973) · Zbl 0284.73002
[52] Noll, W., Continuum mechanics and geometric integration theory, (Lawvere, F. W.; Schanuel, S. H., Categories in Continuum Physics, Lecture Notes in Mathematical Physics, Vol. 1174, (1986), Springer-Verlag Berlin, Heidelberg), 17-29
[53] Noll, W., The geometry of contact, separation, and reformation of continuous bodies, Archive for Rational Mechanics and Analysis, 122, 197-212, (1993) · Zbl 0787.73006
[54] Noll, W.; Seguin, B., Basic concepts of thermomechanics, Journal of Elasticity, 101, 121-151, (2010) · Zbl 1294.80002
[55] Noll, W.; Virga, E., Fit regions and functions of bounded variations, Archive for Rational Mechanics and Analysis, 102, 1-21, (1988) · Zbl 0668.73005
[56] Noll, W.; Virga, E., On edge interactions and surface tension, Archive for Rational Mechanics and Analysis, 111, 1-31, (1990) · Zbl 0709.73001
[57] Simpson, J. A.; Weiner, E. S.C., Oxford English dictionary, (2000), Clarendon Press Oxford
[58] Picado, J., Pultr, A. (2012). Frames and locales, topology without points, Springer Basel. · Zbl 1231.06018
[59] Prusa, V.; Rajagopal, K. R., Jump conditions in stress relaxation and creep experiments of Burgers type fluids: A study in the application of colombeau algebra of generalized functions, Zetischrift fur Angewandte Mathematik und Physik, 62, 707-740, (2011) · Zbl 1292.76007
[60] Rajagopal, K. R., On implicit constitutive theories, Applications of Mathematics, 48, 279-319, (2003) · Zbl 1099.74009
[61] Rajagopal, K. R., Elasticity of elasticity, Zeitschrift fur Angewandte Mathematik und Physik, 58, 309-317, (2007) · Zbl 1113.74006
[62] Rajagopal, K. R., Conspectus of concepts of elasticity, Mathematics and Mechanics of Solids, 16, 536-562, (2011) · Zbl 1269.74014
[63] Rajagopal, K. R.; Walton, J., Modeling fracture in the context of a strain-limiting theory of elasticity: A single anti-plane shear crack, International Journal of Fracture, 169, 39-48, (2011) · Zbl 1283.74074
[64] Robinson, A., Non-standard analysis, (1974), North-Holland Publishing Company Amsterdam-London
[65] (Rose, N., Maxims and minims, (1988), Rome Press Inc North Carolina)
[66] Rubin, M. B., Cosserat theories: shells, rods and points, (2000), Kluwer Academic Publishers Dodrech/Boston/London · Zbl 0984.74003
[67] Russell, B., Analysis of matter, (2007), Spokesman Books Russell House, Nottingham
[68] Russell, B., The principles of mathematics, (1996), W.W. Norton Company New York, London
[69] Russell, B., Portraits from memory and other essays, (1956), Simon and Schuster New York
[70] Shaw, B., Man and superman, (1903), The University Press Cambridge
[71] Sikorski, R., Boolean algebra, (1986), Springer-Verlag Berlin · Zbl 0040.17101
[72] Smith, S. R., Continuous bodies, impenetrability, and contact interactions: the view from the applied mathematics of continuum mechanics, British Journal for the Philosophy of Science, 58, 503-538, (2007) · Zbl 1123.74003
[73] Sorabji, R., Matter, space and motion: theories in antiquity and their sequel, (1988), Cornell University Press New York
[74] Sorabji, R., Time, creation and the continuum, (1983), Cornell University Press Ithaca, New York
[75] (Stewart, M. A., Discourse of things above reason, in selected philosophical papers of Robert Boyle, (1991), Hackett Publishing Company Indianapolis-Cambridge)
[76] Sullivan, T. F., Affine geometry having a solid as primitive, Notre Dame J. Formal Logic, XII, 1-61, (1971) · Zbl 0177.00902
[77] Tarski, A., LES fondements de la geometrie des corps, Ksiega Pamiatkowa Pierwszego Polskiego Zjazdu Matamatecznego Lwow, 7-10, IX, 29-33, (1927)
[78] Tarski, A., Foundation of geometry of solids, (Logic Semantics Metamathematics, (1956), Clarendon Press Oxford), 24-30, [transl. Woodger, J. H.]
[79] Thompson (Lord Kelvin), W., Note on the integration of the equations of equilibrium of an elastic solid, The Cambridge and Dublin Mathematical Journal, 3, 87-89, (1848)
[80] Truesdell, C., The elements of continuum mechanics, (1966), Springer-Verlag New York, Berlin, Heidelberg Tokyo · Zbl 0188.58803
[81] Truesdell, C., A first course in rational continuum mechanics, (1991), Academic Press New York · Zbl 0866.73001
[82] Truesdell, C.; Toupin, R., The classical field theories of mechanics, (Flügge; Siegfried, Principles of classical mechanics and field theory/Prinzipien der Klassischen Mechanik und Feldtheorie, Handbuch der Physik (Encyclopedia of Physics), III/1, (1960), Springer-Verlag Berlin, Heidelberg, New York) · Zbl 0119.19201
[83] Truesdell, C.; Noll, W., The non-linear field theories of mechanics, (Flügge; Siegfried, The non-linear field theories of mechanics/Die Nicht-Linearen Feldtheorien der Mechanik, Handbuch der Physik (Encyclopedia of Physics), III/3, (1965), Springer-Verlag Berlin, Heidelberg, New York) · Zbl 0779.73004
[84] Whitehead, A. N., On the concept of nature, (1920), Cambridge University Press Cambridge · JFM 47.0049.03
[85] Whitehead, A.N. (1979). La Theorie Relationiste de l’Espace, Revue de Metaphysique et de Morale 23, 423-454 (1916). [translated as Hurley, P. J., Relational theory of space, Philosophy Research Archives, 5, 712-741].
[86] Whitehead, A. N.; Russell, B., Principia Mathematica to ^{∗}56, (1962), Cambridge University Press · Zbl 0101.24902
[87] Zimmerman, D. W., Could extended objects be made out of simple parts? an argument for “atomless gunk”, Philosophy and Phenomenological Research, 56, 1-29, (1996)
[88] Zimmerman, D. W., Indivisible parts and extended objects: some philosophical episodes from topology’s prehistory, Monist, 79, 148-180, (1996)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.