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  1. On Abstraction in Mathematics and Indefiniteness in Quantum Mechanics.David Ellerman - 2021 - Journal of Philosophical Logic 50 (4):813-835.
    ion turns equivalence into identity, but there are two ways to do it. Given the equivalence relation of parallelness on lines, the #1 way to turn equivalence into identity by abstraction is to consider equivalence classes of parallel lines. The #2 way is to consider the abstract notion of the direction of parallel lines. This paper developments simple mathematical models of both types of abstraction and shows, for instance, how finite probability theory can be interpreted using #2 abstracts as “superposition (...)
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  • How Category Theory Works.David Ellerman - manuscript
    The purpose of this paper is to show that the dual notions of elements & distinctions are the basic analytical concepts needed to unpack and analyze morphisms, duality, and universal constructions in the Sets, the category of sets and functions. The analysis extends directly to other concrete categories (groups, rings, vector spaces, etc.) where the objects are sets with a certain type of structure and the morphisms are functions that preserve that structure. Then the elements & distinctions-based definitions can be (...)
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  • Toward a General Theory of Knowledge.Luis M. Augusto - 2020 - Journal of Knowledge Structures and Systems 1 (1):63-97.
    For millennia, knowledge has eluded a precise definition. The industrialization of knowledge (IoK) and the associated proliferation of the so-called knowledge communities in the last few decades caused this state of affairs to deteriorate, namely by creating a trio composed of data, knowledge, and information (DIK) that is not unlike the aporia of the trinity in philosophy. This calls for a general theory of knowledge (ToK) that can work as a foundation for a science of knowledge (SoK) and additionally distinguishes (...)
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  • Probability Theory with Superposition Events.David Ellerman - manuscript
    In finite probability theory, events are subsets S⊆U of the outcome set. Subsets can be represented by 1-dimensional column vectors. By extending the representation of events to two dimensional matrices, we can introduce "superposition events." Probabilities are introduced for classical events, superposition events, and their mixtures by using density matrices. Then probabilities for experiments or `measurements' of all these events can be determined in a manner exactly like in quantum mechanics (QM) using density matrices. Moreover the transformation of the density (...)
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