THE XVIII CONFERENCE ON FAMEMS AND THE IV WORKSHOP ON HILBERT’S SIXTH PROBLEM,KRASNOYARSK,SIBERIA,RUSSIA, 2019 The axiom of experience vs the axiom of chance in the theory of experience and chance and co∼eventum mechanics Oleg Yu. Vorobyev Institute of mathematics and computer science Siberian Federal University Krasnoyarsk mailto:oleg.yu.vorobyev@gmail.com http://www.sfu-kras.academia.edu/OlegVorobyev http://olegvorobyev.academia.edu Abstract: This explains the fundamental difference between the two axioms. One of which, the axiom of chance, was introduced by Kolmogorov with his axiomatization of the theory of probability, although he did not call it an axiom. The other, the axiom of experience, which plays a similar role in the dual theory of believability, is formulated in essentially different ways. To facilitate the perception of the meaning of this difference, as well as its far-reaching consequences, which relate to the concept of the past experience and the future chance, these axioms are accompanied by illustrations in the form of Venn diagrams. Keywords: Eventology, probability theory, event, probability, eventological distribution, Gibbs distribution, Boltzmann distribution, hyperbolic distribution, multivariate distribution, entropy, relative entropy, matter, life, mind, Kolmogorov’s axiomatics, believability, certainty, believability theory, theory of experience and chance, co∼event, co∼eventum mechanics, co∼eventum axiomatics, axiom of experience, axiom of chance. MSC: 60A05, 60A10, 60A86, 62A01, 62A86, 62H10, 62H11, 62H12, 68T01, 68T27, 81P05, 81P10, 91B08, 91B10, 91B12, 91B14, 91B30, 91B42, 91B80, 93B07, 94D05 ⟨Ω| |Ω⟩ |x a ⟩ |x b ⟩ ⟨x| |x⟩ ⟨x a | ⟨x b | ⟨x d | |ω⟩ ↑ ⟨x e | |x d ⟩ |x e ⟩ ∀⟨ω|∈⟨x| ↓ : ⟨ω| = ⟨ω| ↓ (∙) ∃! |ω⟩∈|x⟩ ↑ : |ω⟩ = |ω⟩ ↑ (∙) The bra-axiom of a past experience The ket-axiom of a future chance Figure 1: Venn diagrams: the axiom of a past experience (left, in bra-space ⟨Ω|) and the axiom of a future chance (right, in ket- space |Ω⟩). The bra-events ⟨x a | , ⟨x b | , ⟨x d | , ⟨x e |, and so on, that are contained in ⟨x|, are causes of the bra-event ⟨x| that has been experienced/accumulated/corroborated in the past by the observer ⟨x|. The ket-events |x a ⟩ , |x b ⟩ , |x d ⟩ , |x e ⟩, and so on, that contain |x⟩, are consequences of the ket-event |x⟩ that will be possible in the future observations. Here the possible ket-outcomes |ω⟩∈|Ω⟩ and the accumulated bra-outcomes ⟨ω|∈⟨Ω| are shown as (∘), the only ket-outcome |ω⟩ = |ω⟩ ↑ is shown as (∙), and the all bra-outcomes ⟨ω| = ⟨ω| ↓ are shown as (∙). c ○ 2019 O.Yu.Vorobyev This is an open-access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited. Oleg Vorobyev (ed.), Proc. of the XVIII FAMEMS’2019, Krasnoyarsk: Siberian Federal University Press