Philosophy of Mathematics Today

1. Front Matter

   Pages i-xxix

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2. General Philosophical Perspectives

   1. Front Matter

      Pages 1-1

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   2. Logic, Mathematics, Ontology

      • Francisco Miró Quesada

      Pages 3-37

   3. From Certainty to Fallibility in Mathematics?

      • Marco Borga

      Pages 39-50

   4. Moderate Mathematical Fictionism

      • Mario Bunge

      Pages 51-71

   5. Language and Coding-Dependency of Results in Logic and Mathematics

      • Paul Weingartner

      Pages 73-87

   6. What is a Profound Result in Mathematics?

      • Gilles Gaston Granger

      Pages 89-100

   7. The Hylemorphic Schema in Mathematics

      • René Thom

      Pages 101-113

3. Foundational Approaches

   1. Front Matter

      Pages 115-115

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   2. Categorical Foundations of the Protean Character of Mathematics

      • Saunders Mac Lane

      Pages 117-122

   3. Category Theory and Structuralism in Mathematics: Syntactical Considerations

      • Jean-Pierre Marquis

      Pages 123-136

   4. Reflection in Set Theory the Bernays-Levy Axiom System

      • Gert H. Müller

      Pages 137-169

   5. Structuralism and the Concept of Set

      • Charles Parsons

      Pages 171-194

   6. Aspects of Mathematical Experience

      • Wilfried Sieg

      Pages 195-217

   7. Logicism Revisited in the Propositional Fragment of Leśniewski’s Ontology

      • Arata Ishimoto

      Pages 219-232

4. The Applicability of Mathematics

   1. Front Matter

      Pages 233-233

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   2. The Relation of Mathematics to the Other Sciences

      • Evandro Agazzi

      Pages 235-259

   3. Mathematics and Physics

      • Giovanni M. Prosperi

      Pages 261-267

   4. The Mathematical Overdetermination of Physics

      • Erhard Scheibe

      Pages 269-285

   5. Gödel’s Incompleteness Theorem and Quantum Thermodynamic Limits

      • Daniele Mundici

      Pages 287-298

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.

• experience
• logic
• mathematics
• philosophy of mathematics
• physics
• reason
• set theory

• University of Fribourg, Switzerland

  Evandro Agazzi

• University of Genova, Italy

  Evandro Agazzi

• Symmetrion — The Institute for Advanced Symmetry Studies, Budapest, Hungary

  György Darvas

• The Hungarian Academy of Sciences, Budapest, Hungary

  György Darvas

Policies and ethics