Abstract
Can a theory turn back, as it were, upon itselfand vouch for its own features? That is, canthe derived elements of a theory be the veryprimitive terms that provide thepresuppositions of the theory? This form of anall-embracing feature assumes a totality inwhich there occurs quantification over thattotality, quantification that is defined bythis very totality. I argue that the Machprinciple exhibits such a feature ofall-embracing nature. To clarify the argument,I distinguish between on the one handcompleteness and on the other wholeness andtotality, as different all-embracing features:the former being epistemic while the latter –ontological.
I propose an analogy between the Mach principleas a possible selection principle in generalrelativity, and the vicious-circle principle infoundations of mathematics. I finally concludewith a consequence of this analogyvis-à-vis completeness and totality,viz., both should be constrained if they wereto be valid concepts for a physical theory.
The paper progresses chronologically. Itfocuses on the physical approach of Mach thatformed the background for Einstein's generaltheory of relativity. The solutions of thefield equations in the form of cosmologicalmodels set the scene for the view ofall-embracing concepts discussed in the paper.Specifically, the ideas encapsulated in whatEinstein called the Mach principle, constitutethe thread of this account. The principle isfound however to falter, in view of the factthat there are several different types ofsolution of the field equations that contradictit. One such important cosmological model withramifying consequences is the rotational masssolution of Gödel. The question arises asto whether there is an analogy betweenincompleteness in foundations of mathematicsand in physics?
The analogy between the vicious-circleprinciple and the Mach principle demonstratesan affirmative answer which suggests in turnthat completeness and totality must becurtailed – that is, conditions and limitsshould be imposed on completeness and totalityto render them valid for physical theories.
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REFERENCES
Adler, R., M. Bazin and M. Schiffer: 1965, Introduction to General Relativity. New York: McGraw-Hill.
Armour, L.: 1960, The Ontological Argument and the Concepts of Completeness and Selection. The Review of Metaphysics14: 280–291.
Barbour, J.B.: 1989, Absolute or Relative Motion? A Study from a Machian Point of View of the Discovery and the Structure of Dynamical Theories; Vol. 1: The Discovery of Dynamics. Cambridge: Cambridge University Press.
Barbour, J.B.: 1999, The End of Time: The Next Revolution in Our Understanding of the Universe. London: Weidenfeld & Nicolson.
Barbour, J.B. and H. Pfister (eds.): 1995,Mach's Principle: from Newton's Bucket to Quantum Gravity. Einstein's Studies Vol. 6, Boston/Berlin/Basel: Birkhäuser.
Bartels, A.: 2002, Das Gödel-Universum und die Philosphie der Zeit. In Buldt et al. (2002), 231–250.
Barwise, J. and L. Moss: 1996, Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena. Stanford, California: CSLI Publications.
Bergmann, P.G.: 1970, Ernst Mach and Contemporary Physics. In Cohen and Seeger (1970), 69–78.
Bernays, P.: 1967, Scope and Limits of Axiomatics. In M. Bunge (ed.), Delware Seminar in the Foundations of Physics. Berlin/New York/Heidelberg: Springer-Verlag, 188–191.
Bernstein, J. and G. Feinberg (eds.): 1986, Cosmological Constants: Papers in Modern Cosmology. New York: Columbia University Press.
Bradley, J.: 1971, Mach's Philosophy of Science. London.
Bridgman, P.W.: 1961, Significance of the Mach Principle. Am. J. Phys.29: 32–36.
Bub, J.: 1974, The Interpretation of Quantum Mechanics. Dordrecht: Reidel.
Buldt, B., E. Köhler, M. Stöltzner, P. Weibel, C. Klein and W. Depauli-Schimanovich-Göttig (Hrsg.): 2002, Kurt Gödel: Wahrheit und Beweisbarkeit, Band 2. Kompendium zum Werk, Wien: öbv & Hölder-Pichler-Tempsky.
Bulloff, J.J., T.C. Holyoke and S.W. Hahn (eds.): 1969, Foundations of Mathematics: Symposium Papers Commemorating the Sixtieth Birthday of Kurt Gödel. New York: Springer-Verlag.
Bunge, M.: 1966, Mach's Critique of Newtonian Mechanics. Am. J. Phys.34: 585–596.
Carrier, M.: 1994, The Completeness of Scientific Theories. Dordrecht/Boston/London: Kluwer.
Chevalley, C.: 1988, Physical Reality and Closed Theories in W. Heisenberg's Early Papers. In D. Batens and J.P. van Bendegem (eds.), Theory and Experiment. Dordrecht: Reidel, 159–176.
Chihara, C.S.: 1973, Ontology and the Vicious-Circle Principle. Ithaca: Cornell University Press.
Chiu, H.Y. and W.F. Hoffmann (eds.): 1964, Gravitation and Relativity. New York: Benjamin.
Cohen, R.S. and R.J. Seeger (eds.): 1970, ErnstMach: Physicist and Philosopher. Boston Studies in the Philosophy of Science, vol. 6. Dordrecht: Reidel.
Copi, I.M.: 1979, Symbolic Logic, 5th edn. New York: Macmillan.
da Costa, N.C. and F.A. Doria: 1991, Undecidability and Incompleteness in Classical Mechanics. International Journal of Theoretical Physics30: 1041–1073.
De Sitter, W.: 1917, On Einstein's Theory of Gravitation, and Its Astronomical Consequences. Monthly Notices Royal Astronomical Society78, 3–28; in Bernstein and Feinberg (1986), 27–48.
Dicke, R.H.: 1963, Cosmology, Mach's Principle and Relativity. Am. J. Phys.31: 500–509.
Dicke, R.H.: 1964, The Many Faces of Mach and The Significance for the Solar System of Time-Varying Gravitation. In Chiu and Hoffmann (1964), Chs. 7, 8.
Earman, J.: 1989, World Enough and Space-Time. Cambridge, MA: A Bradford Book, MIT Press.
Ehlers, J.: 1995, Machian Ideas and General Relativity. In Barbour and Pfister (1995), 458–469.
Einstein, A.: 1916, Die Grundlage der allgemeinen Relativitätstheorie. Annalen der Physik4th ser. 49: 769ff.
Einstein, A.: 1917, Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie. Sitzungsberichte der Preussischen Akad. d. Wiss. Berlin, 142–152; for an English translation see Bernstein and Feinberg (1986), pp. 16–26; Lorentz et al. (1952), pp. 175–188.
Einstein, A.: 1918a, Prinzipielles zur allgemeinen Relativitätstheorie. Annalen der Physik4th ser. 55: 241–244.
Einstein, A.: 1918b, Kritisches zu einer von Hrn. De Sitter gegebenen Lösung der Gravitationsgleichungen. Sitzungsberichte der Preussischen Akad. d. Wiss. Berlin (7. März 1918), 270–272.
Einstein, A.: 1922, Sidelights on Relativity. London: Methuen.
Einstein, A.: 1955, The Meaning of Relativity. Princeton, New Jersey: Princeton University Press.
Einstein, A.: 1961, Relativity, the Special and the General Theory. New York: Crown Publishers.
Einstein, A., B. Podolsky and N. Rosen: 1935, Can Quantum-Mechanical Description of Physical Reality be Considered Complete. Phys. Rev.47: 777–780.
Ellis, G.F.R.: 1975, Cosmology and Verifiability.Qu. J. Roy. Astro. Soc.16: 245–264.
Ellis, G.F.R.: 1989, The Expanding Universe: a History of Cosmology from 1917 to 1960. In Howard and Stachel (1989), 367ff.
Feyerabend, P.: 1995, Killing Time. Chicago/London: Chicago University Press.
Floyd, J. and S. Shieh (eds.): 2001, Future Pasts: The Analytic Tradition in Twentieth-Century Philosophy. Oxford: Oxford University Press.
Frank, P.: 1970, Einstein, Mach, and Logical Positivism. In Schilpp (1970), 269–286.
Friedman, M.: 1983, Foundations of Space-Time Theories. Princeton, New Jersey: Princeton University Press.
Giere, R.N. and A.W. Richardson (eds.): 1996, Origins of Logical Empiricism. Minnesota Studies in the Philosophy of Science, Vol. XVI. Minneapolis: University of Minnesota Press.
Gödel, K.: 1944, Russell's Mathematical Logic. In P.A. Schilpp (ed.), The Philosophy of Bernard Russell. Chicago: Northwestern University Press, 125–153.
Gödel, K.: 1949a, An Example of a New Type of Cosmological Solutions of Einstein's Field Equations of Gravitation. Rev. Mod. Phys. 21: 447–450.
Gödel, K.: 1949b, A Remark about the Relationship Between Relativity Theory and Idealistic Philosophy. In Schilpp (1970), 557–562.
Gödel, K.: 1950, Rotating Universes in General Relativity Theory. In L.M. Graves, E. Hille, R.A. Smith and O. Zariski (eds.), Proceedings of the 1950 International Congress of Mathematics, vol. I. Providence, RI: American Mathematical Society, 1952, 175–181.
Gödel, K.: 1990, Collected Works. In S. Feferman et al. (eds.), Vol. 2: Publications 1938–1974. Oxford: Oxford University Press.
Gödel, K.: 1995,Collected Works. In S. Feferman et al. (eds.), Vol. 3: Unpublished Essays and Lectures. Oxford: Oxford University Press.
Goenner, H.: 1970, Mach's Principle and Einstein's Theory of Gravitation. In Cohen and Seeger (1970), 200–215.
Goenner, H.: 1995, Mach's Principle and Theories of Gravitation. In Barbour and Pfister (1995), 442–453.
Grim, P.: 1991, The Incomplete Universe: Totality, Knowledge and Truth. Cambridge,MA/London: A Bradford Book, The MIT Press.
Grünbaum, A.: 1974, Philosophical Problems of Space and Time. Boston Studies in the Philosophy of Science, XII. Dordrecht: Reidel.
Hawking, S.: 1990, Introductory Note to 1949and 1952. In Gödel (1990), 189–190.
Hawking, S. and G.F.R. Ellis: 1973, The Large Scale Structure of Space-Time. Cambridge: Cambridge University Press.
Heelan, P.A.: 1965, Quantum Mechanics and Objectivity. The Hague: Martinus Nijhoff.
Hiebert, E.N.: 1980, Ernst Mach. In C.C. Gillispie (ed.), Dictionary of Scientific Biographies, vol. 7. New York: Scribner.
Hintikka, J. (ed.): 1968, A Symposium on Ernst Mach. Synthese18: 132–301.
Hintikka, J.: 2001, Ernst Mach at the Crossroads of Twentieth-Century Philosophy. In Floyd and Shieh (2001), 81–100.
Hoefer, C.: 1994, Einstein's Struggle for a Machian Gravitation Theory. Studies in History and Philosophy of Science25: 287–335.
Holton, G.: 1970, Mach, Einstein, and the Search for Reality. In Cohen and Seeger (1970), 165–199.
Hon, G.: 1987, On Kepler's Awareness of the Problem of Experimental Error. Annals of Science44: 545–591.
Hon, G.: 1995, Is the Identification of Experimental Error Contexturally Dependent? The Case of Kaufmann's Experiment and Its Varied Reception. In J. Buchwald (ed.), Scientific Practice: Theories and Stories of Doing Physics. Chicago, Illinios: Chicago University Press, 170–223.
Hon, G.: 1996, Disturbing, But Not Surprising: Did Gödel Surprise Einstein With a Rotating Universe and Time Travel? Foundations of Physics26: 501–521.
Hooker, C.A.: 1991, Projection, Physical Intelligibility, Objectivity and Completeness: The Divergent Ideals of Bohr and Einstein. Brit. J. Phil. Sci. 42: 491–511.
Howard, D.: 1996, Relativity, Eindeutigkeit, and Monomorphism: Rudolf Carnap and the Development of the Categoricity Concept in Formal Semantics. In Giere and Richardson (1996), 115–164.
Howard, J. and J. Stachel (eds.): 1989, Einstein and the History of General Relativity. Boston: Birkhäuser.
Isenberg, J.: Wheeler-Einstein-Mach Spacetimes. In Barbour and Pfister (1995), 188–203.
Kerszberg, P.: 1989, The Invented Universe: The Einstein-De Sitter Controversy (1916–1917) and the Rise of Relativistic Cosmology. Oxford: Clarendon Press.
Klein, O.: 1962, Mach's Principle and Cosmology in Their Relation to General Relativity. In Recent Developments in General Relativity. Oxford: Pergamon Press, 291–302.
Koslow, A.: 1968, Mach's Concept of Mass: Program and Definition. In Hintikka (1968), 216–233.
Kreisel, G.: 1952, On the Concepts of Completeness and Interpretation of Formal Systems. Fundamenta Mathematica39: 103–127.
Kreisel, G.: 1987, Gödel's Excursions into Intuitionistic Logic. In P. Weingartner and L. Schmetterer (eds.), Gödel Remembered. Napoli: Bibliopolis, 65ff.
Lamb, H.: 1914, Dynamics. Cambridge: Cambridge University Press.
Lanczos, C.: 1972, Einstein's Path from Special to General Relativity. In L. O'Raifeartaigh (ed.), General Relativity, papers in honour of J.L. Synge. Oxford: Clarendon Press, 5–19.
Lorentz, H.A., A. Einstein, H. Minkowski and H. Weyl: 1952, The Principle of Relativity, with notes by A. Sommerfeld. New York: Dover.
Mach, E.: 1974, The Science of Mechanics: A Critical and Historical Account of Its Development, T.J. McCormack (trans.). La Salle, Illinois: Open Court.
Mach, E.: 1988, Die Mechanik in ihrer Entwicklung, Historisch-kritisch dargestellt, Herausgegeben und mit einem Anhang versehen von Renate Wahsner und Horst-Heino von Borzeszkowski. Berlin: Akademie-Verlag.
Malament, D.: 1984, “Time Travel” in the Gödel-universe. Proceedings of the Philosophy of Science Associations2: 91–100.
Malament, D.: 1987, A Note about Closed Timelike Curves in Gödel Space-Time. J. Math. Phys. 28: 2427–2430.
Merleau-Ponty, J.: 1965, Cosmologie du XX e Siècle: étude épistémologique et historique des théories de la cosmologie contemporaine. Paris: Gallimard.
Moon, P. and D.E. Spencer: 1959, Mach's Principle. Philosophy of Science26: 125ff.
Munitz, M.K.: 1962, The Logic of Cosmology. Brit. J. Phi. Sci. 13: 34–50.
Nahin, P.J.: 1999, Time Machines. New York: AIP Press.
Narlikar, J.V.: 1995, Direct Particle Formulation of Mach's Principle. In Barbour and Pfister (1995), 250–261.
Newburgh, R.G.: 1973, Comments on “The Incompatibility of Mach's Principle and the Principle of Equivalence in Current Gravitation Theory”. Brit. J. Phil. Sci.24: 263–264.
Newton, I.: 1969, Mathematical Principles of Natural Philosophy, revised English translation, vol. I. New York: Greenwood Press.
North, J.D.: 1965, The Measure of the Universe. Oxford: Clarendon Press.
Norton, J.: 1983, How Einstein Found His Field Equations: 1912–1915. Historical Studies in the Physical Sciences14: 253–316.
Norton, J.: 1985, What was Einstein's Principle of Equivalence? Studies in History and Philosophy of Science16: 203–246.
Pais, A.: 1982, Subtle is the Lord. Oxford: Oxford University Press.
Paty,M.: 1988, Sur la notion de completude d'une theorie physique. In N. Fleury, S. Joffily, J.A. Martins Simões and A. Troper (eds.), Leite Lopes Festschrift. A pioneer physicist in the third world[ouvrage offert à J. Leite Lopes en l'honneur de son soixante-dixième anniversaire]. Singapore: World Scientific Publishers, 143–164.
Paty,M.: 1993, Einstein philosophe: La physique comme pratique philosophique. Paris: Presses Universitaires de France.
Pfarr, J.: 1981, Time travel in Gödel's Space. General Relativity and Gravitation13: 1073–1091.
Priest, G.: 1995, Beyond the Limits of Thought. Cambridge: Cambridge University Press.
Raine, D.J.: 1981, Mach's Principle and Space-Time Structure. Rep. Prog. Phys.44: 1151–1195.
Ramsey, F.: 1965, The Foundations of Mathematics. London: Routledge & Kegan Paul.
Ray, C.: 1987, The Evolution of Relativity. Bristol: Adam Hilger.
Ray, C.: 1990, The Cosmological Constant: Einstein's Greatest Mistake? Studies in History and Philosophy of Science21: 589–604.
Ray, C.: 1991, Time, Space and Philosophy. London: Routledge.
Rohrlich, F.: 2000, The Validity Limits of Physical Theories: Response to the Preceding Letter. Phys. Lett. A295: 320–322.
Sachs, M.: 1970, Positivism, Realism, and Existentialism in Mach's Influence on Contemporary Physics. Philosophy and Phenomenological Research30: 403–420.
Sachs, M.: 1972, On the Mach Principle and Relative Space-Time. Brit. J. Phil. Sci.23: 117–119.
Sachs, M.: 1974, Ideas of the Theory of Relativity. Jerusalem: Keter.
Sachs, M.: 1975, On the Mach Principle and General Relativity. Brit. J. Phil. Sci.26: 49–51.
Schilpp, P.A. (ed.): 1970, Albert Einstein: Philosopher-Scientist. La Salle, Illinois: Open Court.
Schlegel, R.: 1967, Completeness in Science. New York: Appleton-Century-Crofts.
Sklar, L.: 1995, Philosophy of Physics. Oxford: Oxford University Press.
Smolin, L.: 1997, The life of the Cosmos. New York: Oxford University Press.
Stein, H.: 1970, On the Paradoxical Time-Structures of Gödel. Phil. of Science37: 589–601.
Strauss, M.: 1968, Einstein's Theories and the Critics of Newton. In Hintikka (1968), 251–284.
Stöltzner, M.: 2002, Zeitreisen, Singularitäten und die Unvollständigkeit physikalischer Axiomatik. In Buldt et al. (2002), 289–304.
Wang, H.: 1974, From Mathematics to Philosophy. London: Routledge.
Wang, H.: 1991, Reflections on Kurt Gödel. Cambridge, Massachusetts: The MIT Press, a Bradford Book (3rd printing).
Wheeler, J.A.: 1964, Mach's Principle as Boundary Condition for Einstein's Equations. In Chiu and Hoffmann (1964), 303–349.
Whitehead, A.N. and B. Russell: 1963, Principia Mathematica, 2nd edn. Cambridge: Cambridge University Press.
Wolters, G.: 1987, Mach I, Mach II, Einstein und die Relativitätstheorie. Berlin: de Gruyter.
Woodward, J.F. and W. Yourgrau: 1972, The Incompatibility of Mach's Principle and the Principle of Equivalence in Current Gravitation Theory. Brit. J. Phil. Sci.23: 111–116.
Woodward, J.F. and W. Yourgrau: 1973, Mach's Principle, the Equivalence Principle and Gravitation: A Rejoinder to Newburgh. Brit. J. Phil. Sci.24: 264–270.
Yourgrau, P.: 1991, The Disappearance of Time: Kurt Gödel and the Idealistic Tradition in Philosophy. Cambridge: Cambridge University Press.
Yourgrau, P.: 1999, Gödel meets Einstein: Time Travel in the Gödel Universe. Chicago and La Salle, Illinois: Open Court.
Yourgrau, W.: 1969, Gödel and Physical Theory. Mind78: 77–90.
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Hon, G. Gödel, Einstein, Mach: Casting Constraints on All-embracing Concepts. Foundations of Science 9, 25–64 (2004). https://doi.org/10.1023/B:FODA.0000014880.52117.f0
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DOI: https://doi.org/10.1023/B:FODA.0000014880.52117.f0