Advertisement

Foundations of Science

, Volume 24, Issue 1, pp 1–38 | Cite as

Some Mathematical, Epistemological, and Historical Reflections on the Relationship Between Geometry and Reality, Space–Time Theory and the Geometrization of Theoretical Physics, from Riemann to Weyl and Beyond

  • Luciano BoiEmail author
Article
  • 75 Downloads

Abstract

The history and philosophy of science are destined to play a fundamental role in an epoch marked by a major scientific revolution. This ongoing revolution, principally affecting mathematics and physics, entails a profound upheaval of our conception of space, space–time, and, consequently, of natural laws themselves. Briefly, this revolution can be summarized by the following two trends: (1) by the search for a unified theory of the four fundamental forces of nature, which are known, as of now, as gravity, electromagnetism, and strong and weak nuclear forces; (2) by the search for new mathematical concepts capable of elucidating and therefore explaining such a relationship. In fact, the first search is essentially dependent on the second; that is to say, that in order for a new theory of physics to come to light, the development of a deeper geometric theory capable of explaining the structure of space–time on a quantum scale appears to be necessary. On careful consideration, we notice that both of these developments converge in the direction of a unitary and fundamental tendency of modern science—which is the geometrization of theoretical physics and of natural sciences. This new emergent situation carries within it a profound conceptual change, affecting the way in which relations are conceived of, first and foremost, between mathematics and physics. This new paradigm can be summed up by the intimately interdependent points: (1) the immense variety of physical phenomena and of natural forms follows from the equally infinite variety of geometric and topological objects that can be made out in space and from which space is made up; (2) the second point, which ensues from the former one and which is of great historical and epistemological significance, is that mathematics is involved in rather than applied to phenomena. In other words, phenomena are effects that emerge from the geometrical structure of space–time. There is no doubt that this new conception of the relationship between the universe of mathematical ideas and objects and the world of natural phenomena is the true scientific revolution of our century, of great conceptual importance, and consequently, capable of changing our view of science and of nature at one and the same time. It is all at once of a scientific, philosophical and aesthetic order.

Keywords

Geometry Reality Space Geometrization Unification Strings 

Notes

Acknowledgements

The author was supported by the John Simon Guggenheim Memorial Foundation (New York) and the Canadian Council for Social Sciences and the Humanities (Ottawa), to whom he would like to express his deep gratitude. The author also warmly acknowledges the suggestions, comments and criticisms of Professors Piet Hut, Chiara Nappi and Irving Lavin of the IAS in Princeton.

References

  1. Abraham, R. H., & Shaw, C. D. (1983). Dynamics: The geometry of behavior. Santa Cruz: Aerial Press.Google Scholar
  2. Ageno, M. (1960). Some remarks on the shape of viruses. Nuovo Cimento (Suppl.), 18, 166–175.Google Scholar
  3. Appelquist, T., Chodos, A., & Freund, P. (Eds.). (1987). Modern Kaluza–Klein theories. Reading: Addison-Wesley Publishing Company.Google Scholar
  4. Atiyah, M. (1979). Geometry of Yang–Mills fields. Pisa: Lezioni Fermiane, Accademia Nazionale dei Lincei, Scuola Normale Superiore.Google Scholar
  5. Atiyah, M. (1988). Topological quantum field theories. Institut des Hautes Etudes Scientifiques, Publications Mathématiques, 68, 175–176.Google Scholar
  6. Bachelard, G. (1938). La formation de l’esprit scientifique. Paris: Vrin.Google Scholar
  7. Baez, J., & Muniain, J. P. (1994). Gauge fields, knots and gravity. Singapore: World Scientific.Google Scholar
  8. Bennequin, D. (1994). Questions de physique galoisienne. In Passions des formes. Dynamique qualitative, sémiophysique et intelligibilité, devoted to the work of René Thom, ENS Editions Fontanay St-Cloud (pp. 311–410).Google Scholar
  9. Blum, H. (1974). A geometry for biology. Annals of the New York Academy of Sciences, 231, 19–30.Google Scholar
  10. Bohm, D., & Hiley, B. J. (1993). The undivided universe. London: Routledge.Google Scholar
  11. Bohm, D., & Peat, F. D. (1987). Science, order and creativity. Toronto: Bantan Books.Google Scholar
  12. Boi, L. (1992a). The ‘revolution’ in the geometrical vision of space in the nineteenth century, and the hermeneutical epistemology of mathematics. In D. Gillies (Ed.), Revolutions in mathematics (pp. 183–208). Oxford: Oxford University Press.Google Scholar
  13. Boi, L. (1992b). L’espace: Concept abstrait et/ou physique; la géométrie entre formalisation mathématique et étude de la nature. In L. Boi, D. Flament, J.-M. Salanskis (Eds.), 18301930: A century of geometry. Epistemology, history and mathematics (pp. 63–90). Heidelberg: Springer.Google Scholar
  14. Boi, L. (1994a). Die Beziehungen zwischen Raum, Kontinuum und Materie im Denken Riemanns; die Äthervorstellung und die Einheit der Physik. Das Entstehen einer neuen Naturphilosophie. Philosophia Naturalis, 30(2), 171–216.Google Scholar
  15. Boi, L. (1994b). Mannigfaltigkeit und Gruppenbegriff. Zu den Veränderung der Geometrie in 19. Jahrhundert. Mathematische Semesterberichte, 41(1), 1–16.Google Scholar
  16. Boi, L. (1995a). Le problème mathématique de l’espace. Une quête de l’intelligible, Préface de R. Thom. Heidelberg: Springer.Google Scholar
  17. Boi, L. (1995b). Le concept de variété et la nouvelle géométrie de l’espace dans la pensée de B. Riemann. Archives Internationales d’Histoire des Sciences, 45(134), 82–128.Google Scholar
  18. Boi, L. (1996a). Les géométries non-euclidiennes, le problème philosophique de l’espace et la conception transcendentale; Helmholtz et Kant, les néo-Kantiens, Einstein, Poincaré et Mach. Kant Studien, 87(3), 257–289.Google Scholar
  19. Boi, L. (1996b). Géométries non-euclidiennes, théorie des groupes et conception de l’espace chez Poincaré. In J.-L. Greffe, G. Heinzmann, & K. Lorenz (Eds.), Henri Poincaré—Science and philosophy (pp. 315–332). Berlin/Paris: Akademie Verlag/A. Blanchard.Google Scholar
  20. Boi, L. (1997). La géométrie: Clef du réel? Pensée de l’espace et philosophie des mathématiques. Philosophiques, 24(2), 389–430.Google Scholar
  21. Boi, L. (2000a). Géométrie de l’espace-temps et nature de la physique: Quelques réflexions historiques et épistémologiques. Manuscrito, 23(1), 31–98.Google Scholar
  22. Boi, L. (Ed.). (2000b). Science et Philosophie de la Nature. Un nouveau dialogue. Bern: Peter Lang.Google Scholar
  23. Boi, L. (2004a). Theories of space–time in modern physics. Synthese, 139(3), 429–489.Google Scholar
  24. Boi, L. (2004b). Geometrical and topological foundations of theoretical physics: From gauge theories to string program. International Journal of Mathematics and Mathematical Sciences, 34, 1777–1836.Google Scholar
  25. Boi, L. (2006a). Topological knot theory and macroscopic physics. In J.-P. Françoise, G. Naber, & T. S. Tsun (Eds.), Encyclopedia of mathematical physics (Vol. 5, pp. 271–278). Oxford: Elsevier.Google Scholar
  26. Boi, L. (2006b). From riemannian geometry to Einstein’s general relativity theory and beyond: Space–time structure, geometrization and unification. In J.-M. Alimi & A. Füzfa (Eds.), Proceedings Albert Einstein century international conference (pp. 1066–1075). Melville: American Institute of Physics Publisher.Google Scholar
  27. Boi, L. (2006c). The Aleph of Space. On some extensions of geometrical and topological concepts in the twentieth-century mathematics: From surfaces and manifolds to knots and links. In D. Sica (Ed.), What is Geometry? A special volume of the “Advanced studies in mathematics and logic series” (pp. 79–152). Milan: Polimetrica International Scientific Publisher.Google Scholar
  28. Boi, L. (2008). Topological ideas and structures in fluid dynamics. JP Journal of Geometry and Topology, 8(2), 151–184.Google Scholar
  29. Boi, L. (2009a). Geometria e dinamica dello spazio-tempo nelle teorie fisiche recenti. Giornale di Fisica, 50, 1–10.Google Scholar
  30. Boi, L. (2009b). Ideas of geometrization, geometric invariants of low-dimensional manifolds, and topological quantum field theories. International Journal of Geometric Methods in Modern Physics, 6(5), 701–757.Google Scholar
  31. Boi, L. (2011). The quantum vacuum. A scientific and philosophical concept: From electrodynamics to string theory and the geometry of the microscopic world. Baltimore: The Johns Hopkins University Press.Google Scholar
  32. Boi, L. (2018). Geometry and perception. Mathematical modelling and philosophical interpretation of opatial perception.Google Scholar
  33. Bourguignon, J.-P., & Lawson, H. B. (1982). Yang–Mills theory: Its physical origins and differential geometric aspects. In S.-T. Yau (Ed.), The annals of mathematics studies: Seminar on differential geometry (pp. 395–421). Princeton: Princeton University Press.Google Scholar
  34. Burde, G., & Zieschang, H. (1985). Knots. Berlin-New York: Walter de Gruyter.Google Scholar
  35. Cao, T. Y. (1997). Conceptual developments of 20th century field theories. Cambridge: Cambridge University Press.Google Scholar
  36. Carbone, A., & Semmes, S. (1996). Looking from the inside and from the outside. Preprint de l’Institut des Hautes Études Scientifiques, IHES/M/96/44 (pp. 1–31).Google Scholar
  37. Cartan, E. (1955). Oeuvres complètes (Vol. III, 1). Paris: Gauthier-Villars.Google Scholar
  38. Cassirer, E. (1910). Substanzbegriff und Funktionbegriff. Berlin: Springer.Google Scholar
  39. Chandrasekhar, S. (1987). Truth and Beauty. Aesthetic and Motivations in Science (pp. 66–67). Chicago: The University Chicago Press.Google Scholar
  40. Châtelet, G. (1993). Les enjeux du mobile. Paris: Gallimard.Google Scholar
  41. Chern, S. S. (1989). Selected papers (Vol. III). New York: Springer.Google Scholar
  42. Chern, S. S., & Simons, J. (1974). Characteristic forms and geometrical invariants. Annals of Mathematics, 99, 48–69.Google Scholar
  43. Clifford, W. K. (1879). Lectures and essays (Vol. I). London: Macmillian and Co.Google Scholar
  44. Connes, A. (1994). Noncommutative geometry. New York: Academic.Google Scholar
  45. D’Arcy, T. W. (1942). On growth and form (Vol. 2). Cambridge: Cambridge University Press.Google Scholar
  46. Desanti, J.-T. (1968). Les idéalités mathématiques. Paris: Seuil.Google Scholar
  47. Devine, B., & Wilczek, F. (1988). Longing for the Harmonies. Themes and variations from modern physics. New York: W.W. Norton & Company.Google Scholar
  48. Donaldson, S. K. (1994). Gauge theory and four-manifold topology. In A. Joseph et al. (Eds.), Proceedings first European congress of mathematics (Vol. I, pp. 121–151). Basel: Birkhäuser.Google Scholar
  49. Einstein, A. (1914). Die formale Grundlage der allgemeine Relativitätstheorie (pp. 831–839). Preussische Akademie der Wissenschaften zu Berlin: Sitzungsberichte.Google Scholar
  50. Ellis, G. F. R., & Sciama, D. W. (1972). Global and non global problems in cosmology. In L. O’Raifeartaigh (Ed.), General relativity. Oxford: Oxford University Press.Google Scholar
  51. Flexner, A. (1939). The usefulness of useless knowledge. Harper’s Magazine, October.Google Scholar
  52. Fock, V. A. (1926). Über die invariante Form der Wellen- und der Bewegungsgleichungen für einen geladenen Massenpunkt. Zeitschrift für Physik, 39, 226–232.Google Scholar
  53. Forgacs, P., & Manton, N. S. (1980). Space–time symmetries in Gauge theories. Communications in Mathematical Physics, 72, 15–46.Google Scholar
  54. Freed, D. S., & Uhlenbeck, K. (1991). Instantons and four-manifold. New York: Springer.Google Scholar
  55. Fröhlich, J. (1974). Selected paper. London: World Scientific Publishing.Google Scholar
  56. Gauss, C. F. (1827). Disquisitiones generales circa superfies curvas. In Werke (Göttingen 1873), Vol. IV (pp. 217–258).Google Scholar
  57. Goodwin, B., Kauffman, S. A., & Murray, J. D. (1993). Is morphogenesis an intrinsically robust process. Journal of Theoretical Biology, 162, 135–144.Google Scholar
  58. Gromov, M. (1994). Carnot-Caratheodory spaces seen from within. Preprint IHES, M/94/6.Google Scholar
  59. Holton, G. (1973). Thematic origins of scientific thought: From Kepler to Einstein. Cambridge, MA: Harvard University Press.Google Scholar
  60. Husserl, E. (1913). Ideas pertaining to a pure phenomenology and to a phenomenological philosophy, first book (transl. of the 1th German ed.). The Hague: Martinus Nijhoff Publ., 1982.Google Scholar
  61. Hut, P. (1996). Structuring reality: The role of limits. In J. L. Casti & A. Karlqvist (Eds.), Boundaries and Barriers. On the limits of scientific knowledge (pp. 148–187). New York: Addison-Wesley.Google Scholar
  62. Isham, C. J. (1984). Topological and global aspects of quantum theory. In B. S. DeWitt & R. Stora (Eds.), Relativity, groups and topology II (pp. 1059–1290). Amsterdam: North-Holland.Google Scholar
  63. James, M. (1989). The conceptual development of quantum mechanics. New York: American Institute of Physics.Google Scholar
  64. Kampis, G. (1994). Biological evolution as a process viewed internally. In H. Atmanspacher & G. J. Dalenoort (Eds.), Inside versus outside (pp. 85–110). Berlin: Springer.Google Scholar
  65. Kant, I. (1990). Kritik der reinen Vernunft (1781–1787). Hambourg: Felix Meiner Verlag.Google Scholar
  66. Kauffman, L. H. (Ed.). (1995). Knots and applications. Singapore: World Scientific.Google Scholar
  67. Kepler, J. (1968). Harmonices mundi (first edition in Latin, 1610), Brussels.Google Scholar
  68. Kerszberg, P. (1992). Of exact and inexact sciences in modern physical science. In L. Hardy & L. Embree (Eds.), Phnomenology of natural science. Dordrecht: Kluwer.Google Scholar
  69. Kibble, T. W. B. (1979). Geometrization of quantum mechanics. Communications in Mathematical Physics, 65, 189–201.Google Scholar
  70. Kobayashi, S. (1957). Theory of connections. Annali di Matematica Pura ed Applicata, 43, 119–194.Google Scholar
  71. Kockelmans, J. (1971). On the meaning of scientific revolutions. In R. Gotesky & E. Laszlo (Eds.), Evolution-revolution (pp. 231–252). London: Gordon and Breach.Google Scholar
  72. Koyré, A. (1973). Etudes d’histoire de la pensée scientifique. Paris: Gallimard.Google Scholar
  73. Kuhn, T. (1970). The structure of scientific revolutions. Chicago: University of Chicago Press.Google Scholar
  74. Lachièze-Rey, M., & Luminet, J.-P. (1995). Cosmic topology. Physics Reports, 254, 135–214.Google Scholar
  75. Lambert, D. (1996). Recherches sur la structure et l’efficacité des interactions recentes entre mathématiques et physique. PhD thesis, Université Catholique de Louvain, Institut Supérieur de Philosophie (p. 468). Unpublished.Google Scholar
  76. Largeault, J. (1988). Principes classiques d’interprétation de la nature. Paris: Vrin.Google Scholar
  77. Lautman, A. (1977). Essai sur l’unité des mathématiques. Paris: Union générale d’Editions.Google Scholar
  78. Lee, H. C. (Ed.). (1990). Physics, geometry, and topology. New York: Plenum Press.Google Scholar
  79. Leibniz, G. W. (1992). Philosophischen Schriften, vol. I, edited by C.J. Gerhardt (1875–1890), new edition by H. Herring. Darmstadt: Wissenschaftliche Buchgesellschaft.Google Scholar
  80. Lobatchevsky, N. I. (1829–1830). Geometrical investigations on the theory of parallel lines. On the foundations of geometry (translation from the Russian first edition), Kazan Journal.Google Scholar
  81. London, F. (1927). Quantenmechanische Deutung der Theorie von Weyl. Zeitschrift für Physik, 42, 375–389.Google Scholar
  82. Mach, E. (1906). Erkenntnis und Irrtum. Skizzen zur Psychologie der Forschung: Verlag von J.A. Barth, Leipzig.Google Scholar
  83. Manin, Yu I. (1982). Mathematics and physics. Boston: Birkhäuser.Google Scholar
  84. Manin, Yu I. (1988). Gauge field theory and complex geometry. Heidelberg: Springer.Google Scholar
  85. Mardesic, S., & Segal, J. (Eds) (1987). Geometric topology and shape theory, “Lectures Notes in Mathematics”, No. 1283. Heidelberg: Springer.Google Scholar
  86. Maxwell, J.-C. (1873). Treatise on electricity and magnetism (Vol. 2). Oxford: Clarendon Press.Google Scholar
  87. Miller, A. I. (1984). Imagery in scientific thought. Creating 20th-century physics. Cambridge, MA: MIT Press.Google Scholar
  88. Milnor, J., & Stasheff, J. (1974). Characteristic classes, Annals of Math. Studies, No. 76. Princeton: Princeton University Press.Google Scholar
  89. Misner, C. W., & Wheeler, J. A. (1957). Classical physics as geometry. Annals of Physics, 2, 525–603.Google Scholar
  90. Noether, E. (1918). “Invariante Variationsprobleme”, Nachrichten der Gesellschaft. der Wissenschaften, Göttingen: Math.-Phys. Klasse, 235–257.Google Scholar
  91. O’Raifeartaigh, L. (1997). The dawning of Gauge theory. Princeton: Princeton University Press.Google Scholar
  92. Pauli, W. (1994). Writings on physics and philosophy, Edited by Ch.P. Enz & K. von Meyenn. Heidelberg: Springer.Google Scholar
  93. Penrose, R. (1968). Structure of space–time. In C. M. De Witt & J. A. Wheeler (Eds.), Battelle rencontres (pp. 121–235). New York: W.A. Benjamin.Google Scholar
  94. Penrose, R. (1989). The Emperor’s new mind. Oxford: Oxford University Press.Google Scholar
  95. Poincaré, H. (1902). La science et l’hypothèse. Paris: Flammarion.Google Scholar
  96. Poincaré, H. (1916–1956). Oeuvres (Vols. 1–11). Paris: Gauthier-Villars.Google Scholar
  97. Prismas, H. (1994). Endo- and exo-theories of matter. In H. Atmanspacher & G. J. Dalenoort (Eds.), Inside versus outside (pp. 163–193). Berlin: Springer.Google Scholar
  98. Rashevsky, N. (1956). The geometrization of Biology. Bulletin of Mathematical Biophysics, 18, 31–56.Google Scholar
  99. Regge, T. (1992). Physics and differential geometry. In L. Boi et al. (Eds.), 1830–1930: A century of geometry. Epistemology, history and mathematics (pp. 270–273). Heidelberg: Springer.Google Scholar
  100. Riemann, B. (1854). Über die Hypothesen, welche der Geometrie zu Grunde liegen, Habilitationsschrift, Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Band 13.Google Scholar
  101. Riemann, B. (1990). Gesammelte mathematische Werke, wissenschaftlicher Nachlass und Nachträge/Colleted Papers, new edition edited by R. Narasimhan, Berlin, Leipzig. New York: Springer.Google Scholar
  102. Rovelli, C. (1995). Outline of a generally covariant quantum field theory and a quantum theory of gravity. Journal of Mathematical Physics, 36(11), 6529–6547.Google Scholar
  103. Rushing, T. B. (1973). Topological embeddings. London: Academic.Google Scholar
  104. Salam, A. (1980). Gauge unification of fundamental forces. Reviews of Modern Physics, 92, 525–536.Google Scholar
  105. Saunders, P. T. (1992). The organism as a dynamical system. In F. Varela & W. Stein (Eds.), Thinking about biology. Reading, MA: Addison Wesley.Google Scholar
  106. Scheibe, E. (1982). Invariance and covariance. In J. Agassi & R. S. Cohen (Eds.), Scientific philosophy today, essays in Honor of M. Bunge (pp. 311–331). Dordrecht: D. Reidel.Google Scholar
  107. Schrödinger, E. (1956). The expanding universe. Cambridge: Cambridge University Press.Google Scholar
  108. Schwarz, H. (1982). Superstring theory. Physics Reports, 89, 223–322.Google Scholar
  109. Schwarz, A. S. (1993). Quantum field theory and topology. Berlin: Springer.Google Scholar
  110. Shubnikov, A. V., & Koptsik, V. A. (1974). Symmetry in science and art. New York: Plenum Press.Google Scholar
  111. Singer, I. M. (1987). Some problems in the quantization of Gauge theories and string theories. In Proceedings Hermann Weyl (Vol. 48, pp. 199–218). Providence: Amer. Math. Soc.Google Scholar
  112. Smale, S. (1963). A survey of some recent developments in differential geometry. Bulletin of the American Mathematical Society, 69, 131–145.Google Scholar
  113. Souriau, J.-M. (1992). Physique et Géométrie. In S. Diner, D. Fargue, & G. Lochak (Eds.), La pensée physique contemporaine (pp. 343–364). Paris: Editions A. Fresnel.Google Scholar
  114. Stamatescu, I.-O. (1994). On renormalization in quantum field theory and the structure of space–time. In E. Rudolph & I.-O. Stamatescu (Eds.), Philosophy, mathematics and modern physics—A dialogue (pp. 67–91). Heidelberg: Springer.Google Scholar
  115. Stenrod, N. (1951). The topology of fibre bundles. Princeton: Princeton University Press.Google Scholar
  116. Taubes, C. H. (1982). Self-dual Yang–Mills connections on non-self dual 4-manifolds. Journal of Differential Geometry, 17, 139–170.Google Scholar
  117. Thom, R. (1969). Topological models in biology. Topology, 8, 313–335.Google Scholar
  118. Thom, R. (1983). Paraboles et Catastrophes, Entretiens sur les mathématiques, la science et la philosophie réalisés par G. Giorello et S. Morini. Paris: Flammarion.Google Scholar
  119. Thom, R. (1990). Apologie du logos. Paris: Hachette.Google Scholar
  120. Thuan, T. X. (1998). Le chaos et l’harmonie. La fabrication du réel. Paris: Fayard.Google Scholar
  121. Torretti, R. (1983). Relativity and geometry. Oxford: Pergamon Press.Google Scholar
  122. Torretti, R. (1986). Conceptual reform in scientific revolutions. In R. B. Marcus, G. J. W. Dorn, & P. Weingarten (Eds.), Logic, methodology and philosophy of science (pp. 413–431). Amsterdam: North-Holland.Google Scholar
  123. Trautman, A. (1980). Fiber bundles, gauge fields, and gravitation. In A. Held (Ed.), General relativity and gravitation (Vol. 1, pp. 287–307). New York: Plenum Press.Google Scholar
  124. van Nieuwenhuizen, P. (1984). An Introduction to Simple Supergravity and the Kaluza–Klein program. In B. S. DeWitt & R. Stora (Eds.), Relativity, Groups and Topology II, Les Houches (pp. 823–932). Amsterdam: North-Holland.Google Scholar
  125. Vizgin, V. P. (1994). Unified field theories in the first third of the 20th century. Basel: Birkhäuser.Google Scholar
  126. Weil, A. (1979). De la métaphysique aux mathématiques. In Collected papers (Vol. II). New York: Springer (first published in Sciences, 1960, 52–56).Google Scholar
  127. Weinberg, S. (1980). Conceptual foundations of the unified theory of weak and electromagnetic interactions. Reviews of Modern Physics, 52(3), 515–523.Google Scholar
  128. Weyl, H. (1918). Gravitation und Elektrizität. Sitzungberichte der Königlichen Preussische Akademie der Wissenschaft, Berlin, 26, 465–480.Google Scholar
  129. Weyl, H. (1929). Elektron und gravitation. Zeitschrift für Physik, 56, 330–352.Google Scholar
  130. Weyl, H. (1931). The theory of groups and quantum mechanics. London: Methuen and Co.Google Scholar
  131. Weyl, H. (1949). Philosophy of mathematics and natural sciences. Princeton: Princeton University Press.Google Scholar
  132. Wheeler, J. A. (1962). Geometrodynamics. London: Academic.Google Scholar
  133. Wigner, E. P. (1960). “The unreasonable effectiveness of mathematics in the natural sciences”, Comm. P. Appl. Math., 13 (1). Re-edited in Symmetries and Reflections. Scientific Essays of Eugene P. Wigner, Indiana University Press, Bloomington, 1967.Google Scholar
  134. Witten, E. (1982). Supersymmetry and Morse theory. Journal of Differential Geometry, 17, 661–692.Google Scholar
  135. Witten, E. (1987). Physics and geometry. In Proceedings of the international congress of mathematics (Berkeley 1986) (pp. 267–303). American Mathematical Society.Google Scholar
  136. Witten, E. (1988). Topological quantum field theory. Communications in Mathematical Physics, 117, 353–386.Google Scholar
  137. Yang, C. N. (1989). Hermann Weyl’s contribution to physics. In K. Chandrasekharan (Ed.), Hermann Weyl centenary lectures (pp. 7–21). Heidelberg: Springer.Google Scholar
  138. Yang, C. N., & Mills, R. L. (1954). Conservation of isotopic-spin and isotopic gauge invariance. Physical Review, 96(1), 191–195.Google Scholar
  139. Yu, T. T., & Yang, C. N. (1975). Concept of non integrable phase factors and global formulation of gauge fields. Physical Review D, 12, 3845–3857.Google Scholar
  140. Zee, A. (Ed.). (1982). Unity of forces in the universe (Vol. I). Singapore: World Scientific.Google Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Centre de Mathématiques and Programme Philosophie et ÉpistémologieEcole des Hautes Etudes en Sciences SocialesParisFrance

Personalised recommendations