Images and Logic of the Light Cone: Tracking Robb’s Postulational Turn in Physical Geometry
DOI:
https://doi.org/10.22370/rhv2016iss8pp43-105Keywords:
Robb, axiomatics, postulates, postulationism, light cone, relativity theory, geometry, foundations of mathematics, space-time, model, logical model, Russell, Hilbert, Veblen, Huntington, Peano, Minkowski, Cambridge, GöttingenAbstract
Previous discussions of Robb’s work on space and time have offered a philosophical focus on causal interpretations of relativity theory or a historical focus on his use of non-Euclidean geometry, or else ignored altogether in discussions of relativity at Cambridge. In this paper I focus on how Robb’s work made contact with those same foundational developments in mathematics and with their applications. This contact with applications of new mathematical logic at Göttingen and Cambridge explains the transition from his electron research to his treatment of relativity in 1911 and finally to the axiomatic presentation in 1914 in terms of postulates. At the heart of Robb’s physical optics was the model of the light cone. The model underwent a transition from a working mechanical model in the Maxwellian Cambridge sense of a pedagogical and research tool to the semantic model, in the logical, model-theoretic sense. Robb tracked this transition from the 19th- to the 20th-century conception with the earliest use of the term ‘model’ in the new sense. I place his cone models in a genealogy of similar models and use their evolution to track how Robb’s physical researches were informed by his interest in geometry, logic and the foundations of mathematics.
References
Barrow-Green, J. & Gray, J.J. (2006). ‘Geometry at Cambridge, 1863-1940’, Historia Mathematica 33, 315-356.
Campbell, N.R. (1910a). ‘The principles of dynamics’, Philosophical Magazine Ser. 6, 19, 168-81.
Campbell, N.R. (1911a). ‘The common sense of relativity’, Philosophical Magazine Ser. 6, 21, 502-517.
Campbell, N.R. (1919). Physics. The Elements, Cambridge: Cambridge University Press.
Carmichael, R.D. (1912). ‘On the theory of relativity: Analysis of the postulates’, Physical Review 35, n.3, 153-176.
Carmichael, R.D. (1913). The Theory of Relativity, New York: John Wiley & Sons.
Cat, J. (2001). ‘On understanding: Maxwell on the methods of illustration and scientific metaphor’, Studies in the History and Philosophy of Modern Physics 33B, n.3, 395-442.
Cat, J. (2016). Intellectual Trajectories and Local Interactions: Carmichael and Davis from Mathematics to Relativity and Philosophy at Indiana University in the 1910s and 20s, Indiana University ms.
Corcoran, J. (1980). ‘On definitional equivalence and related topics’, History and Philosophy of Logic 1, 231-4.
Corry, L. (1997). ‘Hermann Minkowski and the postulate of relativity’, Archive for History of Exact Sciences 51, n.4, 273-314.
Corry, L. (2004). David Hilbert and the Axiomatization of Physics (1898-1918), Dordrecht: Kluwer.
Cunningham, E.C. (1914). ‘Letter to the editor: The Principle of Relativity’, Nature 92, 2331, 454.
Darrigol, O. (2000). Electrodynamics from Ampère to Einstein, Chicago: University of Chicago Press.
Darrigol, O. (2014). Physics and Necessity, Oxford: Oxford University Press.
DiSalle, R. (1993). ‘Helmholtz’s empiricist philosophy of mathematics: Between laws of perception and laws of nature’, in D. Cahan, ed., Hermann von Helmholtz and the Foundations of Nineteenth-Century Science, Berkeley: University of California Press, 498-521.
Ferreirós, J. (2006). ‘Riemann’s Habilitationsvortrag at the Crossroads at Mathematics, Physics and Philosophy’, in J. Ferreirós and J.J. Gray, eds., The Architecture of Modern Mathematics: Essays in History and Philosophy, Oxford: Oxford University Press, 67-96.
Galison, P.L. (1979). ‘Minkowski’s space-time: from visual thinking to the absolute world’, Historical Studies in the Physical Sciences 10, 85-121.
Goldberg, S. (1984). Understanding Relativity: Origin and Impact of a Scientific Revolution, Boston: Birkhäuser.
Hilbert, D. (1899). Grundlagen der Geometrie, Leipzig: Teubner.
Hobson, E.W. & Love, A.E.H. (eds.) (1913). Proceedings of the Fifth International Congress of Mathematicians, 2 vols., Cambridge: Cambridge University Press.
Hunt, B.J. (1991). The Maxwellians, Ithaca, NY: Cornell University Press.
Huntington, E.V. (1902). ‘A complete set of postulates for the theory of absolute continuous magnitude’, Transactions of the American Mathematical Society 3, n.2, 264-79.
Huntington, E.V. (1911). ‘The fundamental propositions of algebra’, in J.W.A. Young, ed., Monographs on the Topics of Modern Mathematics, New York: Longmans, Green and Co., 151-210.
Huntington, E.V. (1912). ‘A new approach to the theory of relativity’, Philosophical Magazine 23, n.136, 494-513.
Kox, A.J. (1997). ‘The discovery of the electron: II. The Zeeman effect’, European Journal of Physics 18, 139-44.
Larmor, J. (1900). Aether and Matter, Cambridge: Cambridge University Press.
Laub, J. (1910). ‘Über die experimentellen Grundlagen des Relativitätsprinzips’, Jahrbuch der Radioaktivität und Elektronik 7, 405-463.
von Laue, M. (1911). Das Relativitätsprinzip, Braunschweig: Vieweg.
Lennes, N.J. (1911). ‘Review: Edward V. Huntington, The Fundamental Laws of Addition and Multiplication in Elementary Algebra’, Bulletin of the American Mathematical Society 17, n.7, 362-5.
Lewis, G.N. & Tolman, R.C. (1909). ‘The Principle of Relativity, and non-Newtonian mechanics’, Proceedings of the American Academy of Arts and Sciences 44, n. 25, 711-24.
Lorentz, H.A. (1916). The Theory of Electrons and Its Applications to the Phenomena of Light and Radiant Heat, Leipzig: B.G. Teubner.
Mancosu, P. (2003). ‘The Russellian influence on Hilbert and his school’, Synthese 137, 59-101.
Miller, A.I. (1981). Albert Einstein’s Special Theory of Relativit: emergence (1905) and Early Interpretation (1905-1911), New York: Springer.
Minkowski, H. (1909/1911). ‘Raum und Zeit’, Physikalische Zeitschrift 10, 104-111, in Gesammelte Abhandlungen, ed. D. Hilbert, 1911, vol. 2, 431-444.
Minkowski, H. (1915). ‘Das Relativitätsprinzip’, Annalen der Physik 47, 927-38.
Moore, E.H. (1902). ‘On the projective axioms of geometry’, Transactions of the American Mathematical Society 3, 142-158.
von Neumann, J. (1925). ‘Eine Axiomatisierung der Mengenlehre’, Journal für die reine und angewandte Mathematik 22, 219-40.
Padoa, A. (1904). ‘Un nuovo sistema di definizioni par la geometria euclidea’, Periodico di Matematische 1, 74-80.
Peano, G. (1889). I Principii di Geometria Logicamente Esposti, Torino: Fratelli Bocca Editori.
Peano, G. (1894). ‘Sui fondamenti della geometria’, Rivista di Matematica 4, 51-90.
Poincaré, H. (1902). La Science et l’Hypothèse, Paris: Flammarion.
Pressland, A.J. (1892). ‘On the history and degree of certain geometrical approximation’, Proceedings of the Edinburgh Mathematical Society 10, 27.
Pyenson, L. (1976). ‘Einstein’s early scientific collaboration’, Historical Studies in the Physical Sciences 7, 83-123.
Pyenson, L. (1979). ‘Physics in the shadow of mathematics: The Göttingen electron-theory seminar of 1905’, Archive for History of the Exact Sciences 21, n.1, 55-89.
Robb, A.A. (1904). Beiträge zur Theorie des Zeemaneffektes, Leipzig: Johann Ambrosius Barth.
Robb, A.A. (1905a). ‘On the conduction of electricity through gases between parallel plates. Part I’, Philosophical Magazine Ser. 6, 10, n.56, 237-42.
Robb, A.A. (1905b). ‘On the conduction of electricity through gases between parallel plates. Part II’, Philosophical Magazine Ser. 6, 10, n.60, 664-76.
Robb, A.A. (1906). ‘The revolution of the corpuscle’, Nature 1892, n. 73, 321.
Robb, A.A. (1911). Optical Geometry of Motion. A New View of the Theory of Relativity, Cambridge: W. Heffer and Sons.
Robb, A.A. (1913a). ‘Proof of one of Peano’s axioms of the straight line’, The Messenger of Mathematics 42, 121-3.
Robb, A.A. (1913b). ‘Note on the proof of one of Peano’s axioms of the straight line’, The Messenger of Mathematics 42, 134.
Robb, A.A. (1914a). ‘Letter to the editor: The Principle of Relativity’, Nature 92, 2331, 454.
Robb, A.A. (1914b). A Theory of Time and Space, Cambridge: Cambridge University Press.
Robb, A.A. (1921). The Absolute Relations of Time and Space, Cambridge: Cambridge University Press.
Routh, E.J. (1892). An Advanced Treatise on the Dynamics of Rigid Bodies, 2 vols., Cambridge: Cambridge University Press.
Russell, B. (1899). ‘Sur les axiomes de la géométrie’, Revue de Métaphysique et de Morale 7, 684-707.
Russell, B. (1903). Principles of Mathematics, Cambridge: Cambridge University Press.
Russell, B. (1905a). ‘Review of Henri Poincaré’s Science and Hypothesis’, Mind 14, 412-418.
Russell, B., (1905b). ‘On Denoting’, Mind 56, 479-493.
Russell, B. (1914). ‘The relation of sense-data to physics’, Scientia 16, 1-27.
Sánchez-Ron, J.M. (1987). ‘The reception of Special Relativity in Great Britain’, in T. Glick, ed., The Comparative Reception of Relativity, Dordrecht: Reidel, 27-58.
Scanlan, M. (1991). ‘Who were the American postulate theorists?’, Journal of Symbolic Logic 56, n.3, 981-1002.
Schlimm, D. (2003). ‘Axioms in mathematical practice’, Philosophia Mathematica (III) 21, 37-92.
Sommerfeld, A. (1904). ‘Zur elektronentheorie. II. Grundlagen für eine allgemeine Dynamik des Elektrons’, Nachrichten v.d. Königlichen Gesellschaft der Wissenschaften z. Göttingen 1904, n.1, 363-439.
Sommerfeld, A. (1905). Zur Elektronentheorie. III. Ueber Lichtgeschwindigkeits- und Ueberlichtgeschwindigkeits-Elektronen’, Nachrichten v.d. Königlichen Gesellschaft der Wissenschaften z. Göttingen 1905, n.3, 201-235.
Sommerfeld, A. (1910a). ‘Zur Relativitätstheorie. I. Vierdimensionale Vektoralgebra’, Annalen der Physik 337, n. 9, 749-76.
Sommerfeld, A. (1910b). ‘Zur Relativitätstheorie. I. Vierdimensionale Vektoranalysis’, Annalen der Physik 338, n.14, 649-689.
Tarski, A. (1936/1956). ‘On the concept of logical consequence’, Logic, Semantics, Metamathematics, Oxford: Clarendon Press, 409-20.
Tarski, A. (1936/1986). ‘Über den Begriff der logischen Folgerung’, in S.R. Givant and R.N. McKenzie, eds., Alfred Tarski. Collected Papers, vol. 2, 1935-1944, Boston: Birkhäuser, 269-82.
Thomson, J.J. (ed.) (1910). A History of the Cavendish Laboratory 1870-1910, Cambridge: Cambridge University Press.
Tolman, R.C. (1910). ‘The second postulate of relativity’, Physical Review Ser. 1, 31, n.1, 26-40.
Torretti, R. (1978). Philosophy of Geometry from Riemann to Poincaré, Dordrecht: Reidel.
Torretti, R. (1983/1996). Relativity and Geometry, New York: Dover.
Veblen, O. (1903). ‘Hilbert’s foundations of geometry’, The Monist 13, n.2, 303-9.
Veblen, O. (1904). ‘A system of axioms for geometry’, Transactions of the American Mathematical Society 5, n.3, 343-384.
Veblen, O. (1911). ‘The foundations of geometry’, in J.W.A. Young, ed., Monographs on the Topics of Modern Mathematics, New York: Longmans, Green and Co., 3-54.
Veblen, O. & Young, J.W.A. (1910). Projective Geometry, 2 vols., Boston: Ginn and Co.
Voigt, W. (1887). ‘Ueber das Doppler’sche Princip,’ Nachrichten v.d. Königlichen Gesellschaft der Wissenschaften u.d. Georg-August-Universität z. Göttingen 1887, n.2, 41-51.
Voigt, W. (1896). Kompendium der theoretische Physik, 2 vols., Leipzig: Verlag von Veit & Comp.
Voigt, W. (1899). ‘Weiters zur Theorie der Zeemaneffectes’, Annalen der Physik 68, 352-364.
Voigt, W. (1901). Elementare Mechanik, Leipzig: Verlag von Veit & Comp.
Voigt, W. (1908). Magneto- und Elektrooptik Leipzig: B.G. Teubner.
Walter, S. (1999). ‘The non-Euclidean style of Minkowskian relativity’, in J.J. Gray, ed., 1999, The Symbolic Universe, Oxford: Oxford University Press, 99-127.
Warwick, A.C. (2003). Masters of Theory: Cambridge and the Rise of Mathematical Physics, Chicago: University of Chicago Press.
Whitehead, A.N. (1906). The Axioms of Projective Geometry, Cambridge: Cambridge University Press.
Whitehead, A.N. (1907). The Axioms of Descriptive Geometry, Cambridge: Cambridge University Press.
Whitehead, A.N. & Russell, B. (1910-1913). Principia Mathematica, 3 vols., Cambridge: Cambridge University Press.
Windred, G. (1933). ‘The history of mathematical time: II’, Isis 20, n.1, 192-219.
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication, with the work after publication simultaneously licensed under a Creative Commons Attribution License (CC BY-NC-ND 4.0 International) that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
