Skip to main content

Introduction: Leibniz, Turing, Zuse, and Beyond

  • Chapter
  • First Online:
The Universe as Automaton

Part of the book series: SpringerBriefs in Complexity ((BRIEFSCOMPLEXITY))

  • 1360 Accesses

Abstract

According to Einstein, a scientific explanation should be as simple as possible, but not too simple, for it to be realistic. It would be nice to understand the great scientific problems of the universe (such as cosmic expansion, black holes, the evolution of life, and brains) with just basic knowledge. The toy world of cellular automata is an intuitive, but mathematically precise model that may be used to illustrate fundamental problems of topical research. The philosopher and mathematician Gottfried Wilhelm Leibniz (1646–1716), who constructed one of the first mechanical calculating machines, considered, even then, the universe as an automaton created by God as a divine engineer and mathematician. The theory of cellular automata was independently initiated by several computer pioneers, among them John von Neumann (1903–1957) and Konrad Zuse (1910–1995).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 49.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 64.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • H.D.I. Abarbanel, Analysis of Observed Chaotic Data (Springer, New York, 1995)

    MATH  Google Scholar 

  • S. Albeverio, V. Jentsch, H. Kantz (eds.), Extreme Events in Nature and Society (Springer, Berlin, 2006)

    Google Scholar 

  • K.T. Alligood, T.D. Sauer, J.A. Yorke, Chaos: An Introduction to Dynamical Systems (Springer, New York, 1996)

    MATH  Google Scholar 

  • V.I. Arnold, Small denominators II, proof of a theorem of A.N. Kolmogorov on the preservation of conditionally-periodic motions under a small perturbation of the Hamiltonian. Russ. Math. Surveys 18, 5 (1963)

    ADS  Google Scholar 

  • J. Audretsch, K. Mainzer (Hrsg.) Wieviele Leben hat Schrödingers Katze? Zur Physik und Philosophie der Quantenmechanik, 2nd edn. (Spektrum Akademischer Verlag, Heidelberg, 1996)

    Google Scholar 

  • H.-J. Bungartz, S. Zimmer, M. Buchholz, D. Pflüger, Modellbildung und Simulation. Eine anwendungsorientierte Einführung (Springer, Berlin, 2009)

    MATH  Google Scholar 

  • A.W. Burks (ed.), Cellular Automata (University of Illinois Press, Urbana, 1970)

    MATH  Google Scholar 

  • L.O. Chua, CNN: A Paradigm for Complexity (World Scientific, Singapore, 1998)

    Book  MATH  Google Scholar 

  • A. Church, A note on the Entscheidungsproblem. J. Symb. Log. 1, 40–41 (1936)

    Article  MATH  Google Scholar 

  • S. Feferman (ed.), Kurt Gödel: Collected Works (Oxford University Press, Oxford, 1986), pp. 144–195

    Google Scholar 

  • R. Feynman, Simulating physics with computers. Int. J. Theor. Phys. 21(6–7), 467–488 (1982)

    Article  MathSciNet  Google Scholar 

  • E. Fredkin, Digital Mechanics: An informational process based on reversible universal CA. Physica D 45:254–270 (1990)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • R. Gandy, C.E.M. Yates (eds.) Collected Works of A. M. Turing (North-Holland, Amsterdam 1992–2001)

    Google Scholar 

  • M. Gardner, The fantastic combinations of John Conway’s new solitaire game of life. Sci. Am. 223, 120–123 (1970)

    Article  Google Scholar 

  • M. Gardner, Mathematical games: on cellular automata, self-reproduction, the Garden of Eden, and the game “Life”. Sci. Am. 224, 112–117 (1971)

    Article  Google Scholar 

  • R. Herken (ed.) The Universal Turing Machine. A Half-Century Survey, 2nd edn. (Springer, Wien 1995)

    MATH  Google Scholar 

  • A.G. Hoekstra, J. Kroc, P.M.A. Sloot (eds.), Simulating Complex Systems by Cellular Automata (Springer, Berlin, 2010)

    MATH  Google Scholar 

  • A.N. Kolmogorov, On conservation of conditionally-periodic motions for a small change in Hamilton’s function. Dokl. Akad. Nauk. USSR 98, 525 (1954)

    MathSciNet  Google Scholar 

  • K. Mainzer, ComputerNeue Flügel des Geistes? (De Gruyter, Berlin 1994)

    Google Scholar 

  • K. Mainzer, Computerphilosophie (Junius, Hamburg, 2003)

    Google Scholar 

  • K. Mainzer, Symmetry and Complexity: The Spirit and Beauty of Nonlinear Science (World Scientific, Singapore, 2005)

    Book  MATH  Google Scholar 

  • K. Mainzer, Thinking in Complexity. The Computational Dynamics of Matter, Mind, and Mankind, 5th edn. (Springer, Berlin, 2007a)

    MATH  Google Scholar 

  • K. Mainzer, Der kreative Zufall. Wie das Neue in die Welt kommt (C.H. Beck, München, 2007b)

    Google Scholar 

  • K. Mainzer, Komplexität (UTB-Profile, Paderborn, 2008)

    Google Scholar 

  • K. Mainzer (ed.), Complexity. Eur. Rev. (Academia Europaea) 17(2), 219–452 (2009)

    Google Scholar 

  • K. Mainzer, Leben als Maschine? Von der Systembiologie zur Robotik und Künstlichen Intelligenz (Mentis, Paderborn, 2010)

    Google Scholar 

  • K. Mainzer, Mathematischer Konstruktivismus. Dissertation Universität Münster (1973)

    Google Scholar 

  • M. Minsky, Size and structure of universal Turing machines using tag systems, recursive function theory. Proceedings Symposium in Pure Mathematics, vol. 5 (American Mathematical Society, Providence RI), pp. 229–238 (1962)

    Google Scholar 

  • J. Moser, Convergent series expansions of quasi-periodic motions. Math. Ann. 169, 163 (1967)

    Article  Google Scholar 

  • J.V. Neumann, Theory of Self-Reproducing Automata (University of Illinois Press, Urbana, 1966)

    Google Scholar 

  • P. Petrov, Church-Turing thesis as an immature form of Zuse-Fredkin thesis. (More arguments in favour of the ‘universe as a cellular automaton’ idea). 3rd WSEAS International Conference on Systems Theory and Scientific Computation. Special session on cellular automata and applications (2003) (online version: http://digitalphysics.org/Publications/Petrov/Pet02a2/Pet02a2.htm)

  • H. Poincaré, Les Méthodes Nouvelles de la Méchanique Célèste I-II (Gauther-Villars, Paris, 1897)

    Google Scholar 

  • J. Schmidhuber, A computer scientist’s view of life, the universe, and everything. Lecture Notes in Computer Science (Springer, Berlin, 1997), pp. 201–208

    Google Scholar 

  • C. Shannon, in A Universal Turing Machine With Two Internal States. Automata Studies (Princeton University Press, Princeton, 1956), pp. 157–165

    Google Scholar 

  • M. Small, Applied Nonlinear Time Series Analysis: Applications in Physics, Physiology and Finance (World Scientific, Singapore, 2005)

    Book  MATH  Google Scholar 

  • T. Toffoli, N. Margolus, Cellular Automata Machines: A New Environment for Modeling (MIT Press, Cambridge, 1987)

    Google Scholar 

  • B.A. Toole, Ada, the Enchantress of Numbers (Strawberry Press, Mill Valley, CA, 1992)

    Google Scholar 

  • A.M. Turing, Computability and λ-definability. J. Symb. Log. 2, 153–163 (1937)

    Article  MATH  Google Scholar 

  • A.M. Turing, Systems of logic based on ordinals. Proc. Lond. Math. Soc. 2(45), 161–228 (1939)

    Article  Google Scholar 

  • A.M. Turing, The chemical basis of morphogenesis. Philos. Trans. Royal Soc. Lond. Ser. B, Biol. Sci. 237(641), 37–72 (1952)

    Article  ADS  Google Scholar 

  • A.M. Turing, On computable numbers with an application to the Entscheidungsproblem. Proc. Lond. Math. Soc. 42(2) 230–265, corrections, ibid, 43, 544–546 (1936/1937)

    Google Scholar 

  • N. Wiener, A. Rosenbluth, The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle. Arch. Inst. Cardiol. Mex. 16, 205–265 (1946)

    Google Scholar 

  • S. Wolfram, Theory and Applications of Cellular Automata (World Scientific, Singapore, 1986)

    MATH  Google Scholar 

  • S. Wolfram, Cellular Automata and Complexity (Addison–Wesley, Reading, 1994)

    MATH  Google Scholar 

  • S. Wolfram, A New Kind of Science (Wolfram Media, Champaign, 2002)

    MATH  Google Scholar 

  • R. Wright, Three Scientists and Their Gods: Looking for Meaning in an Age of Information (HarperCollins, New York, 1989)

    Google Scholar 

  • K. Zuse, Rechnender Raum (Friedrich Vieweg & Sohn, Braunschweig, 1969)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Klaus Mainzer .

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Klaus Mainzer

About this chapter

Cite this chapter

Mainzer, K., Chua, L. (2012). Introduction: Leibniz, Turing, Zuse, and Beyond. In: The Universe as Automaton. SpringerBriefs in Complexity. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23477-4_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-23477-4_1

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-23476-7

  • Online ISBN: 978-3-642-23477-4

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics