Mathesis Universalis, Computability and Proof
Editors: Centrone, S., Negri, S., Sarikaya, D., Schuster, P.M. (Eds.)
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In a fragment entitled Elementa Nova Matheseos Universalis (1683?) Leibniz writes “the mathesis […] shall deliver the method through which things that are conceivable can be exactly determined”; in another fragment he takes the mathesis to be “the science of all things that are conceivable.” Leibniz considers all mathematical disciplines as branches of the mathesis and conceives the mathesis as a general science of forms applicable not only to magnitudes but to every object that exists in our imagination, i.e. that is possible at least in principle. As a general science of forms the mathesis investigates possible relations between “arbitrary objects” (“objets quelconques”). It is an abstract theory of combinations and relations among objects whatsoever.
In 1810 the mathematician and philosopher Bernard Bolzano published a booklet entitled Contributions to a BetterGrounded Presentation of Mathematics. There is, according to him, a certain objective connection among the truths that are germane to a certain homogeneous field of objects: some truths are the “reasons” (“Gründe”) of others, and the latter are “consequences” (“Folgen”) of the former. The reasonconsequence relation seems to be the counterpart of causality at the level of a relation between true propositions. Arigorous proof is characterized in this context as a proof that shows the reason of the proposition that is to be proven. Requirements imposed on rigorous proofs seem to anticipate normalization results in current proof theory.
The contributors of Mathesis Universalis, Computability and Proof, leading experts in the fields of computer science, mathematics, logic and philosophy, show the evolution of these and related ideas exploring topics in proof theory, computability theory, intuitionistic logic, constructivism and reverse mathematics, delving deeply into a contextual examination of the relationship between mathematical rigor and demands for simplification.
 About the authors

Stefania Centrone is currently Privatdozentin at the University of Hamburg, teaches and does research at the Universities of Oldenburg and of Helsinki and has been in 2016 deputy Professor of Theoretical Philosophy at the University of Göttingen. In 2012 she was awarded a DFGEigene Stelle for the project Bolzanos und Husserls Weiterentwicklung von Leibnizens Ideen zur Mathesis Universalis and 2017 a Heisenberg grant. She is author of the volumes Logic and philosophy of Mathematics in the Early Husserl (Synthese Library 2010) and Studien zu Bolzano (Academia Verlag 2015).
Sara Negri is Professor of Theoretical Philosophy at the University of Helsinki, where she has been a Docent of Logic since 1998. After a PhD in Mathematics in 1996 at the University of Padova and research visits at the University of Amsterdam and Chalmers, she has been a research associate at the Imperial College in London, a Humboldt Fellow in Munich, and a visiting scientist at the MittagLeffler Institute in Stockholm. Her research interests range from mathematical logic and philosophy of mathematics to proof theory and its applications to philosophical logic and formal epistemology.
Deniz Sarikaya is PhDStudent of Philosophy and studies Mathematics at the University of Hamburg with experience abroad at the Universiteit van Amsterdam and Universidad de Barcelona. He stayed a term as a Visiting Student Researcher at the University of California, Berkeley developing a project on the Philosophy of Mathematical Practice concerning the Philosophical impact of the usage of automatic theorem prover and as a RISE research intern at the University of British Columbia. He is mainly focusing on philosophy of mathematics and logic.
Peter Schuster is Associate Professor for Mathematical Logic at the University of Verona. After both doctorate and habilitation in mathematics at the University of Munich he was Lecturer at the University of Leeds and member of the Leeds Logic Group. Apart from constructive mathematics at large, his principal research interests are about the computational content of classical proofs in abstract algebra and related fields in which maximum or minimum principles are invoked.  Reviews

 Table of contents (19 chapters)


Introduction: Mathesis Universalis, Proof and Computation
Pages 16

Diplomacy of Trust in the European Crisis: Contributions by the Alexander von Humboldt Foundation
Pages 711

Mathesis Universalis and Homotopy Type Theory
Pages 1336

Note on the Benefit of Proof Representations by Name
Pages 3745

Constructive Proofs of Negated Statements
Pages 4753

Table of contents (19 chapters)
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Bibliographic Information
 Bibliographic Information

 Book Title
 Mathesis Universalis, Computability and Proof
 Editors

 Stefania Centrone
 Sara Negri
 Deniz Sarikaya
 Peter M. Schuster
 Series Title
 Synthese Library
 Series Volume
 412
 Copyright
 2019
 Publisher
 Springer International Publishing
 Copyright Holder
 Springer Nature Switzerland AG
 eBook ISBN
 9783030204471
 DOI
 10.1007/9783030204471
 Hardcover ISBN
 9783030204464
 Edition Number
 1
 Number of Pages
 X, 374
 Number of Illustrations
 38 b/w illustrations
 Topics