# Spurious, Emergent Laws in Number Worlds

^{1}

^{2}

^{*}

## 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 |

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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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**MDPI and ACS Style**

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