Abstract. The 20th century and the beginning of the 21st century were marked by the
deontologisation of the language of science, the dominance of the attitudes of epistemological naturalism, the loss of the truth’s absolute value and the purpose of cognition, and as a consequence, the unprecedented increase in the relevance of rethinking mathematics and mathematical natural science (physics, biology, etc.) in terms of everyday consciousness and everyday experience. This work is connected with the necessity to clarify the boundaries ofidentity and differences between the formations of sensory-visual experience and the products of abstract-theoretical construction, the intuitive and the logical-discursive, explanation and understanding, the theoretical and the empirical, etc., in the context of the analysis of the deriving oppositions and counter-views as expressions of opposites of reflexive and non-reflexive “layers” of consciousness in structures of cognitive and vital-practical activity.
Keywords: ordinary consciousness, ontology, epistemology, crisis in the foundation of
mathematics, revolution in physics, constructivism, logicism, formalism, intuitionism,
conventionalism, linguistic turn, scientific rationality
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