# Spurious, Emergent Laws in Number Worlds

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Physical/Universal/Natural Laws

## 3. Laws and Limit Constructions

## 4. Order within Disordered Sequences

## 5. The Emergence of Turing Complete (Universal) Computation

## 6. Is the World Number Computable?

## 7. Non-Uniform Evolution

## 8. Is the Universe Lawless?

## 9. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Causation and Correlation: Two Formal Models

x | y | $\mathit{x}\succ \mathit{y}$ | C(x, y) |
---|---|---|---|

0 | 0 | 0 | 1 |

0 | 1 | 0 | 0 |

1 | 0 | 0 | 0 |

1 | 1 | 1 | 1 |

## References

- Huffman, C. Pythagoreanism. In The Stanford Encyclopedia of Philosophy; Zalta, E.N., Ed.; Metaphysics Research Lab Stanford University: Stanford, CA, USA, 2016. [Google Scholar]
- Maziarz, E.A.; Greenwood, T. Greek Mathematical Philosophy; Ungar: New York, NY, USA, 1968. [Google Scholar]
- Schrödinger, E. Nature and the Greeks; Cambridge University Press: Cambridge, UK, 1954 2014. [Google Scholar]
- Fredkin, E. Digital mechanics. An informational process based on reversible universal cellular automata. Physica
**1990**, D45, 254–270. [Google Scholar] - Margolus, N. Physics-like model of computation. Physica
**1984**, D10, 81–95. [Google Scholar] - Toffoli, T. The role of the observer in uniform systems. In Applied General Systems Research: Recent Developments and Trends; Klir, G.J., Ed.; Plenum Press, Springer: New York, NY, USA; London, UK, 1978; pp. 395–400. [Google Scholar]
- Wolfram, S. A New Kind of Science; Wolfram Media, Inc.: Champaign, IL, USA, 2002. [Google Scholar]
- Zuse, K. Rechnender Raum; Friedrich Vieweg & Sohn: Braunschweig, Germany, 1969. [Google Scholar]
- Zuse, K. Calculating Space; MIT Technical Translation AZT-70-164-GEMIT; MIT (Proj. MAC): Cambridge, MA, USA, 1970. [Google Scholar]
- Tegmark, M. The mathematical universe. Found. Phys.
**2007**, 38, 101–150. [Google Scholar] [CrossRef] - Tegmark, M. Our Mathematical Universe: My Quest for the Ultimate Nature of Reality; Penguin Random House LLC.: New York, NY, USA, 2014. [Google Scholar]
- Calude, C.S.; Meyerstein, W.F.; Salomaa, A. The Universe is Lawless. In Computable Universe: Understanding and Exploring Nature as Computation; World Scientific: Singapore, 2013; pp. 527–537. [Google Scholar]
- Svozil, K. Extrinsic-intrinsic concept and complementarity. In Inside versus Outside; Atmanspacher, H., Dalenoort, G.J., Eds.; Springer Series in Synergetics: Berlin, Germany, 1994; pp. 273–288. [Google Scholar]
- Borges, J.L. La biblioteca de Babel—The Library of Babel; Editorial Sur: Buenos Aires, Argentina, 1941. (in Spanish), 1962 (in English). See also Borges’ 1939 essay “The Total Library” [Google Scholar]
- Anderson, P.W. More is different. Broken symmetry and the nature of the hierarchical structure of science. Science
**1972**, 177, 393–396. [Google Scholar] [CrossRef] [PubMed] - O’Connor, T.; Wong, H.Y. Emergent properties. In The Stanford Encyclopedia of Philosophy; Zalta, E.N., Ed.; Metaphysics Research Lab Stanford University: Stanford, CA, USA, 2015. [Google Scholar]
- Kivelson, S.; Kivelson, S.A. Defining emergence in physics. Npj Quant. Mater.
**2016**, 1, 16024. [Google Scholar] [CrossRef] - Hiebert, E.N. Common frontiers of the exact sciences and the humanities. Phys. Perspect.
**2000**, 2, 6–29. [Google Scholar] [CrossRef] - Stöltzner, M. Vienna indeterminism: Mach, Boltzmann, Exner. Synthese
**1999**, 119, 85–111. [Google Scholar] [CrossRef] - Schweidler, E.v. Über Schwankungen der radioaktiven Umwandlung; H. Dunod & E. Pinat: Paris, France, 1906; pages German part, 1–3. [Google Scholar]
- Exner, F.S. Über Gesetze in Naturwissenschaft und Humanistik: Inaugurationsrede gehalten am 15. Oktober 1908; Hölder, Ebooks on Demand Universitätsbibliothek: Viena, Austria, 1909. [Google Scholar]
- Schrödinger, E. Was ist ein Naturgesetz? Naturwissenschaften
**1929**, 17, 01. [Google Scholar] [CrossRef] - Schrödinger, E. Science And The Human Temperament; George Allen & Unwin: Crows Nest, UK, 1935. [Google Scholar]
- Born, M. Zur Quantenmechanik der Stoßvorgänge. Zeitschrift Phys.
**1926**, 37, 863–867. [Google Scholar] [CrossRef] - Hofstadter, D.R. Artificial Intelligence: Subcognition as Xomputation; Technical Report TR132; Indiana University: Bloomington, IN, USA, 1982. [Google Scholar]
- Forrest, S. Emergent computation: Self-organizing, collective, and cooperative phenomena in natural and artificial computing networks: Introduction to the proceedings of the ninth annual cnls conference. Phys. D Nonlinear Phenomena
**1990**, 42, 1–11. [Google Scholar] [CrossRef] - Hume, D. The Clarendon Edition of the Works of David Hume. In A Treatise of Human Nature: Volume 1: Texts; Clarendon Press and Oxford University Press: Oxford, UK, 2007. [Google Scholar]
- Beebee, P. Hume and the problem of causation. In The Oxford Handbook of Hume; Russell, P., Ed.; Oxford Handbooks; Oxford University Press: Oxford, UK; New York, NY, USA, 2016. [Google Scholar]
- De Pierris, G.; Friedman, M. Kant and Hume on causality. In The Stanford Encyclopedia of Philosophy; Zalta, E.N., Ed.; Metaphysics Research Lab, Stanford University: Stanford, CA, USA, 2013. [Google Scholar]
- Feynman, R.P. The Character of Physical Law; MIT Press: Cambridge, MA, USA, 1965. [Google Scholar]
- Mumford, S.; Anjum, R.L. Causation: A Very Short Introduction; Oxford University Press: Oxford, UK, 2014. [Google Scholar]
- Norton, J.D. Causation as folk science. Philos. Imprint
**2003**, 3, 1–22. [Google Scholar] - Russell, B.I. –On the notion of cause. Proc. Aristotelian Soc.
**1913**, 13, 06. [Google Scholar] [CrossRef] - Armstrong, D.M. What is a Law of Nature? In Cambridge Studies in Philosophy; Cambridge University Press: Cambridge, UK, 1983. [Google Scholar]
- Cabello, A. Physical origin of quantum nonlocality and contextuality. arXiv, 2019; arXiv:1801.06347. [Google Scholar]
- Calude, C.S.; Meyerstein, W.F. Is the universe lawful? Chaos Solit. Fract.
**1999**, 10, 1075–1084. [Google Scholar] - Mueller, M.P. Could the physical world be emergent instead of fundamental, and why should we ask? (short version). arXiv, 2017; arXiv:1712.01826. [Google Scholar]
- Rosen, J. Lawless Universe: Science and the Hunt for Reality; The John Hopkins University Press: Baltimore, MA, USA, 2010. [Google Scholar]
- van Fraassen, B.C. Laws and Symmetry; Oxford University Press: Oxford, UK, 1989 2003. [Google Scholar]
- Yanofsky, N. Chaos makes the multiverse unnecessary. Nautilus
**2017**, 66, 1–16. [Google Scholar] - Rajaraman, A.; Ullman, J.D. Mining of Massive Datasets; Cambridge University Press: Cambridge, UK, 2011. [Google Scholar]
- Reed, D.A.; Dongarra, J. Exascale computing and big data. Commun. ACM
**2015**, 58, 56–68. [Google Scholar] [CrossRef] - Schutt, R.; O’Neil, C. Doing Data Science; O’Reilly Media: Newton, MA, USA, 2014. [Google Scholar]
- Stanton, J.M. Introduction to Data Science; Syracuse University: Syracuse, NY, USA, 2012. [Google Scholar]
- Ellis, G.; Silk, J. Scientific method: Defend the integrity of physics. Nature
**2014**, 516, 321–323. [Google Scholar] [CrossRef][Green Version] - Calude, C.S.; Longo, G. The deluge of spurious correlations in big data. Found. Sci.
**2017**, 22, 595–612. [Google Scholar] [CrossRef] - Hesiod. Hesiod: Volume I, Theogony. Works and Days. Testimonia (Loeb Classical Library No. 57); Harvard University Press: Cambridge, MA, USA; London, UK, 2006. [Google Scholar]
- Curd, P.; McKirahan, R.D. A Presocratics Reader (Second Edition). Selected Fragments and Testimonia; Hackett Publishing Co.: Indianapolis, IN, USA; Cambridge, UK, 2011. [Google Scholar]
- Diels, H.; Kranz, W. Die Fragmente der Vorsokratiker, 6th ed.; Weidmannsche Buchhandlung: Berlin, Germany, 1952. [Google Scholar]
- Kirk, G.S.; Raven, J.E.; Schofield, M. The Presocratic Philosophers: A Critical History with a Selcetion of Texts, 2nd ed.; Cambridge University Press: Cambridge, UK, 1983. [Google Scholar]
- Corbett, R.J. The question of natural law in Aristotle. History Political Thought
**2009**, 30, 229–250. [Google Scholar] - Drake, S. Galileo and the law of inertia. Am. J. Phys.
**1964**, 32, 601–608. [Google Scholar] [CrossRef] - Douven, I. Abduction. In The Stanford Encyclopedia of Philosophy; Zalta, E.N., Ed.; Metaphysics Research Lab Stanford University: Stanford, CA, USA, 2017. [Google Scholar]
- Josephson, J.R.; Josephson, S.G. Abductive Inference: Computation, Philosophy, Technology; Cambridge University Press: Cambridge, UK; New York, NY, USA; Melbourne, Australia, 1994. [Google Scholar]
- Peirce, C.S.; Hartshorne, C.; Weiss, P.; Burks, A.W. Collected Papers of Charles Sanders Peirce; Harvard University Press Belknap Press: Cambridge, MA, USA, 1932. [Google Scholar]
- Laskar, J. A numerical experiment on the chaotic behaviour of the solar system. Nature
**1989**, 338, 237–238. [Google Scholar] [CrossRef] - Noether, E. Invariante Variationsprobleme. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse
**1918**, 1918, 235–257. [Google Scholar] - Popper, K.R. The Logic of Scientific Discovery; Hutchinson & Co and Routledge: New York, NY, USA, 1959 1992. [Google Scholar]
- Gruska, J. Foundations of Computing; International Thompson Computer Press: London, UK, 1997. [Google Scholar]
- Calude, C.S.; Dumitrescu, M. A probabilistic anytime algorithm for the halting problem. Computability
**2018**, 7, 259–271. [Google Scholar] [CrossRef] - Frank, M.; Vieri, C.; Ammer, J.; Love, N.; Margolus, N.H.; Knight, T. A scalable reversible computer in silicon. In Unconventional Models of Computation; Springer: Singapore, 1998; pp. 183–200. [Google Scholar]
- Landauer, R. Computation: A fundamental physical view. Phys. Scr.
**1987**, 35, 88. [Google Scholar] [CrossRef] - Mermin, D.N. Quantum Computer Science; Cambridge University Press: Cambridge, UK, 2007. [Google Scholar]
- Bostwick, C.W.; Rainwater, J.; Baum, J.D. E1321. Am. Math. Mon.
**1959**, 66, 141–142. [Google Scholar] [CrossRef] - Greenwood, R.E.; Gleason, A.M. Combinatorial relations and chromatic graphs. Can. J. Math.
**1955**, 7, 1–7. [Google Scholar] [CrossRef] - Graham, R. Some of my favorite problems in Ramsey Theory. INTEGERS Electronic J. Combinat. Number Theory
**2007**, 7, #A2. [Google Scholar] - Calude, C. Information and Randomness—An Algorithmic Perspective, 2nd ed.; Springer: Berlin, Germany, 2002. [Google Scholar]
- Downey, R.G.; Hirschfeldt, D.R. Algorithmic Randomness and Complexity. In Theory and Applications of Computability; Springer: Berlin, Germany, 2010. [Google Scholar][Green Version]
- Goodman, A.W. On sets of acquaintances and strangers at any party. Am. Math. Mon.
**1959**, 66, 11. [Google Scholar] [CrossRef] - Schwenk, A.J. Acquaintance graph party problem. Am. Math. Mon.
**1972**, 79, 12. [Google Scholar] [CrossRef] - Pawliuk, M.; Waddell, M.A. Using Ramsey Theory to Measure Unavoidable Spurious Correlations in Big Data. Axioms
**2019**, 8, 29. [Google Scholar] [CrossRef] - Chaitin, G.J. Algorithmic Information Theory (Cambridge Tracts in Theoretical Computer Science), revised ed.; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Berlekamp, E.; Conway, J.H.; Guy, R. Winning Ways; Academic Press: New York, NY, USA, 1982. [Google Scholar]
- Rendell, P. Turing Machine Universality of the Game of Life; Springer International Publishing: Cham, Switzerland, 2016. [Google Scholar]
- Abbott, A.A.; Calude, C.S.; Svozil, K. Value indefiniteness is almost everywhere. Phys. Rev. A
**2014**, 89, 032109–032116. [Google Scholar] [CrossRef] - Champernowne, D.G. The construction of decimals normal in the scale of ten. J. Lond. Math. Soc.
**1933**, 8, 254–260. [Google Scholar] [CrossRef] - Calude, C.S.; Chaitin, G.J. What is … a halting probability? Notices AMS
**2007**, 57, 236–237. [Google Scholar] - Calude, C.S.; Calude, E. The complexity of mathematical problems: An overview of results and open problems. Int. J. Unconv. Comput.
**2013**, 9, 327–343. [Google Scholar] - Calude, C. Borel normality and algorithmic randomness. In Developments in Language Theory; Rozenberg, G., Salomaa, A., Eds.; World Scientific: Singapore, 1994; pp. 113–129. [Google Scholar]
- Egan, G. Permutation City; Millennium Orion Publishing Group: London, UK, 1994. [Google Scholar]
- Heidegger, M. Was ist Metaphysik? Klostermann: Frankfurt, Germany, 1929. [Google Scholar]
- Heidegger, M. Einführung in die Metaphysik (Freiburger Vorlesung Sommersemester 1935). In Martin Heidegger Gesamtausgabe; Klostermann: Frankfurt, Germany, 1935. [Google Scholar]
- Jaynes, E.T. Probability Theory: The Logic Of Science; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Pattee, H.H. Postscript: Unsolved Problems and Potential Applications of Hierarchy Theory; Springer: Dordrecht, The Netherlands, 2012; pp. 111–124. [Google Scholar]
- Buiatti, M.; Longo, G. Randomness and multilevel interactions in biology. Theory Biosci.
**2013**, 132, 139–158. [Google Scholar] [CrossRef] [PubMed][Green Version] - Heidelberger, M. Functional relations and causality in Fechner and Mach. Philos. Psychol.
**2010**, 04, 201023. [Google Scholar] [CrossRef] - Spurious Correlations. Available online: http://www.tylervigen.com/spurious-correlations (accessed on 9 April 2019).
- Vigen, T. Spurious Correlations; Hachette Books: New York, NY, USA, 2015. [Google Scholar]

1. | A sequence is infinite while a string is finite. A finite prefix of a sequence is then a string. |

2. | This physical-mathematical mapping assumption is essential for this paper. |

3. | “Emergence is a notorious philosophical term of art.” [16]. In this paper we will not use the term in the sense of the philosophical emergency theory, but with the signification given in physics [17]: “The term emergent is used to evoke collective behaviour of a large number of microscopic constituents that is qualitatively different than the behaviours of the individual constituents." |

4. | See the Appendix A for a more formal discussion. |

5. | |

6. | These limits can be mitigated from a practical point of view with various methods; for example, the halting problem can be solved probabilistically with arbitrarily high precision [60]. |

7. | In fact, there is a second trio who are either mutually acquainted or unacquainted [64]. |

8. | If we interpret 0 and 1 as colours, then the theorem says that in every binary sequence there exist arbitrarily long monochromatic arithmetical progressions. |

9. | Again, the proof is not constructive. |

10. | The finite version of Van der Waerden theorem shows that the same phenomenon appears in long enough strings. See more in [46]. |

11. | A Turing machine with a prefix-free domain is called self-delimiting. A (self-delimiting) Turing machine which can simulate any other (self-delimiting) Turing machine is called universal. A sequence is Martin-Löf random if there exists a fixed constant such that every finite prefix (string) of the sequence cannot be compressed by a self-delimiting universal Turing machine by more than a constant [67]. |

12. | This holds true even constructively. |

13. | Probability zero is not the same as impossibility: there exist infinitely many sequences—like the computable ones—which contain no spurious correlations. |

14. | The minimum length of an input a Turing machine needs to compute a string of length n lies in the interval $(0,n+c)$, where c is a fixed constant. From this it follows that $\alpha \in (0,1)$. |

15. | More precisely, when $n\ge 2/\alpha $. |

16. | A model of computation is Turing complete—sometimes called universal—if it can simulate a universal Turing machine. |

17. | Again, one should not think that this means that there are no computable world numbers, see Section 6. The result follows from the fact that the computable sequences form a countable set. |

18. | A sequence is bi-immune if its corresponding set of natural numbers nor its complement contain an infinite computably enumerable subset. |

19. | In base 10, ${C}_{10}=12345678910111213141516\dots $. |

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Calude, C.S.; Svozil, K. Spurious, Emergent Laws in Number Worlds. *Philosophies* **2019**, *4*, 17.
https://doi.org/10.3390/philosophies4020017

**AMA Style**

Calude CS, Svozil K. Spurious, Emergent Laws in Number Worlds. *Philosophies*. 2019; 4(2):17.
https://doi.org/10.3390/philosophies4020017

**Chicago/Turabian Style**

Calude, Cristian S., and Karl Svozil. 2019. "Spurious, Emergent Laws in Number Worlds" *Philosophies* 4, no. 2: 17.
https://doi.org/10.3390/philosophies4020017