Ebook: Philosophy of Mathematics Today

Mathematics is often considered as a body of knowledge that is essen­ 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 Mathe­ 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 language­ dependency of verisimilitude; 3) The proof of the weak and strong anti­ 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 language­ 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 proposi­ tions and theories.

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The book is of interest to general philosophers of science, thanks to the attention paid to logical, linguistic and ontological issues regarding mathematics. People interested in foundational research will find penetrating papers regarding structuralist and set-theoretical approaches. Scientists may appreciate the analyses of the role of mathematics in several sciences.

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The book is of interest to general philosophers of science, thanks to the attention paid to logical, linguistic and ontological issues regarding mathematics. People interested in foundational research will find penetrating papers regarding structuralist and set-theoretical approaches. Scientists may appreciate the analyses of the role of mathematics in several sciences.
Content:
Front Matter....Pages i-xxix
Front Matter....Pages 1-1
Logic, Mathematics, Ontology....Pages 3-37
From Certainty to Fallibility in Mathematics?....Pages 39-50
Moderate Mathematical Fictionism....Pages 51-71
Language and Coding-Dependency of Results in Logic and Mathematics....Pages 73-87
What is a Profound Result in Mathematics?....Pages 89-100
The Hylemorphic Schema in Mathematics....Pages 101-113
Front Matter....Pages 115-115
Categorical Foundations of the Protean Character of Mathematics....Pages 117-122
Category Theory and Structuralism in Mathematics: Syntactical Considerations....Pages 123-136
Reflection in Set Theory the Bernays-Levy Axiom System....Pages 137-169
Structuralism and the Concept of Set....Pages 171-194
Aspects of Mathematical Experience....Pages 195-217
Logicism Revisited in the Propositional Fragment of Le?niewski’s Ontology....Pages 219-232
Front Matter....Pages 233-233
The Relation of Mathematics to the Other Sciences....Pages 235-259
Mathematics and Physics....Pages 261-267
The Mathematical Overdetermination of Physics....Pages 269-285
G?del’s Incompleteness Theorem and Quantum Thermodynamic Limits....Pages 287-298
Mathematical Models in Biology....Pages 299-304
The Natural Numbers as a Universal Library....Pages 305-317
Mathematical Symmetry Principles in the Scientific World View....Pages 319-334
Front Matter....Pages 335-335
Mathematics and Logics Hungarian Traditions and the Philosophy of Non-Classical Logic....Pages 337-351
....Pages 353-361