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Full-Text Articles in Physical Sciences and Mathematics
Logic Of Scientific Discovery: How Physical Induction Affects What Is Computable, Vladik Kreinovich, Olga Kosheleva
Logic Of Scientific Discovery: How Physical Induction Affects What Is Computable, Vladik Kreinovich, Olga Kosheleva
Departmental Technical Reports (CS)
Most of our knowledge about a physical world comes from physical induction: if a hypothesis is confirmed by a sufficient number of observations, we conclude that this hypothesis is universally true. We show that a natural formalization of this property affects what is computable when processing measurement and observation results, and we explain how this formalization is related to Kolmogorov complexity and randomness. We also consider computational consequences of an alternative idea also coming form physics: that no physical law is absolutely true, that every physical law will sooner or later need to be corrected. It turns out that this …
Deep Mathematical Results Are The Ones That Connect Seemingly Unrelated Areas: Towards A Formal Proof Of Gian-Carlo Rota's Thesis, Olga Kosheleva, Vladik Kreinovich
Deep Mathematical Results Are The Ones That Connect Seemingly Unrelated Areas: Towards A Formal Proof Of Gian-Carlo Rota's Thesis, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
When is a mathematical result deep? At first glance, the answer to this question is subjective: what is deep for one mathematician may not sound that deep for another. A renowned mathematician Gian-Carlo Rota expressed an opinion that the notion of deepness is more objective that we may think: namely, that a mathematical statement is deep if and only if it connects two seemingly unrelated areas of mathematics. In this paper, we formalize this thesis, and show that in this formalization, Gian Carlo Rota's thesis becomes a provable mathematical result.