Deduction, Computation, Experiment Exploring the Effectiveness of Pr
What is a proof for? What is the characteristic use of a proof as a computation, as opposed to its use as an experiment? What is the relationship between mathematical procedures and natural processes? The essays collected in this volume address such quest
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Rossella Lupacchini, Giovanna Corsi (Eds.)
Deduction, Computation, Experiment Exploring the Effectiveness of Proof
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R OSSELLA LUPACCHINI Gianpaolo Carlino Giancarlo D’Ambrosio G IOVANNA CORSI Leonardo Merola Dipartimento di Filosofia Pierluigi degli Paolucci Università Studi di Bologna Giulia Ricciardi Università degli Studi di Napoli “Federico II” and INFN Sezione di Napoli
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Preface
This volume is located in a cross-disciplinary field bringing together mathematics, logic, natural science and philosophy. Reflection on the effectiveness of proof brings out a number of questions that have always been latent in the informal understanding of the subject. What makes a symbolic construction significant? What makes an assumption reasonable? What makes a proof reliable? G¨odel, Church and Turing, in different ways, achieve a deep understanding of the notion of effective calculability involved in the nature of proof. Turing’s work in particular provides a “precise and unquestionably adequate” definition of the general notion of a formal system in terms of a machine with a finite number of parts. On the other hand, Eugene Wigner refers to the unreasonable effectiveness of mathematics in the natural sciences as a miracle. Where should the boundary be traced between mathematical procedures and physical processes? What is the characteristic use of a proof as a computation, as opposed to its use as an experiment? Wha
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