Some Personal Remarks
Abstract
Mathematics is neither the contemplation of an independently given realm nor a mere creation of the mind. It arises from passive and active experience, yet grows through the exploration of hypothetical states of affairs: structures partly constituted by assumptions, representations, symbols, and methods, whose consequences can nevertheless be discovered. This standpoint calls into question sharp separations between pure and applied mathematics, since mathematical concepts and techniques have developed in close interaction with natural knowledge, scientific modelling, and human practices. It also shifts the philosophical focus from the ontology of mathematical objects to the objectivity of mathematical discourse, understood as a historically situated but intersubjectively valid form of conceptual work. Particular attention is given to the role of semiotic systems, notation, diagrams, formulae, and practices in the growth of mathematical knowledge, and to the irreducible complementarity between formal manipulation and conceptual meaning. Foundational questions concerning logic, set theory, continuity, and the continuum problem are approached from the same perspective: not as demands for a single ultimate framework, but as reasons for adopting a pluralistic view of mathematical foundations and of the role of mathematics within science.
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