### Abstract

The question "What is life," has long been discussed, and many concepts representing what are regarded as essential properties of life have been proposed, namely information generating processes, complex systems, indefinite boundaries, and self- reference. Self-referential forms in particular play a central role in autonomous life, because both in ontogenetic and phylogenetic processes we find self-referential forms and self-reference introduces other important features of life. However, because self-reference is strongly connected with self-contradiction and/or unprogrammability, it cannot be described in the paradigm of prediction or dynamics. It is inevitable to estimate the problem of instability of the description itself. We here discuss self-reference and unprogrammability with respect to the mixture between inter- and intracellular computations in biological systems, and we replace the difference between computation velocities at different levels by the difference between the velocity of observation propagation and that of a particle. We also define the formal system of communication and transplant the problem of unprogrammability resulting from the finite velocity of observation propagation in that system. Here it is illustrated that unprogrammability is isomorphic to the incompleteness of a formal system. Further, we formally estimate the relationship of unprogrammability and one-to-many type mapping and propose the formalization using a universal arrow, based on Wittgenstein's idea, language games, and performativeness. Because unprogrammability does not exist as a real entity, but is constituted, we can formalize one-to-many type mapping and even unstable formal description. The concept of autonomy is not beyond formal description; however it is beyond the paradigm of prediction.

Original language | English |
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Pages (from-to) | 19-76 |

Number of pages | 58 |

Journal | Applied Mathematics and Computation |

Volume | 58 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1993 Sep 15 |

Externally published | Yes |

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics