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TitlePhilosophy of Mathematics Today [electronic resource] / edited by Evandro Agazzi, Gyรถrgy Darvas
ImprintDordrecht : Springer Netherlands : Imprint: Springer, 1997
Connect tohttp://dx.doi.org/10.1007/978-94-011-5690-5
Descript XXIX, 361 p. online resource

SUMMARY

Mathematics is often considered as a body of knowledge that is essenยญ tially independent of linguistic formulations, in the sense that, once the content of this knowledge has been grasped, there remains only the problem of professional ability, that of clearly formulating and correctly proving it. However, the question is not so simple, and P. Weingartner's paper (Language and Coding-Dependency of Results in Logic and Matheยญ matics) deals with some results in logic and mathematics which reveal that certain notions are in general not invariant with respect to different choices of language and of coding processes. Five example are given: 1) The validity of axioms and rules of classical propositional logic depend on the interpretation of sentential variables; 2) The languageยญ dependency of verisimilitude; 3) The proof of the weak and strong antiยญ inductivist theorems in Popper's theory of inductive support is not invariant with respect to limitative criteria put on classical logic; 4) The language-dependency of the concept of provability; 5) The languageยญ dependency of the existence of ungrounded and paradoxical sentences (in the sense of Kripke). The requirements of logical rigour and consistency are not the only criteria for the acceptance and appreciation of mathematical proposiยญ tions and theories


CONTENT

General Philosophical Perspectives -- Logic, Mathematics, Ontology -- From Certainty to Fallibility in Mathematics? -- Moderate Mathematical Fictionism -- Language and Coding-Dependency of Results in Logic and Mathematics -- What is a Profound Result in Mathematics? -- The Hylemorphic Schema in Mathematics -- Foundational Approaches -- Categorical Foundations of the Protean Character of Mathematics -- Category Theory and Structuralism in Mathematics: Syntactical Considerations -- Reflection in Set Theory. The Bernays-Levy Axiom System -- Structuralism and the Concept of Set -- Aspects of Mathematical Experience -- Logicism Revisited in the Propositional Fragment of Le?niewskiโ{128}{153}s Ontology -- The Applicability of Mathematics -- The Relation of Mathematics to the Other Sciences -- Mathematics and Physics -- The Mathematical Overdetermination of Physics -- Gรถdelโ{128}{153}s Incompleteness Theorem and Quantum Thermodynamic Limits -- Mathematical Models in Biology -- The Natural Numbers as a Universal Library -- Mathematical Symmetry Principles in the Scientific World View -- Historical Considerations -- Mathematics and Logics. Hungarian Traditions and the Philosophy of Non-Classical Logic -- Umfangslogik, Inhaltslogik, Theorematic Reasoning


Mathematics Epistemology Logic Philosophy and science Mathematical logic Mathematics Mathematical Logic and Foundations Logic Philosophy of Science Epistemology



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