Editors:
Gottfried Leibniz's philosophy of logic in the Digital Age
Views on secondorder thinking in the history of mathematics
A contemporary perspective on the foundations of mathematics
Part of the book series: Synthese Library (SYLI, volume 412)
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Table of contents (19 chapters)

Front Matter
About this book
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. A rigorous 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.Keywords
 Gottfried Wilhelm Leibniz
 Mathesis universalis
 Philosophy of Mathematics
 Proof Theory logic
 Ordinal Analysis
 Characteristica universalis
 Calculus Ratiocinator
 mathesis
 Constructive Mathematics
 Foundations of Mathematics
 Craig's interpolation theorem
 intensional type theory
 Turing Machine Philosophy
 Concept of Mathematics and Classification
 Analytic Philosophy mathematics
 History of Mathematics philosophy
 Reverse mathematics
 Bolzano philosophy
 Metamathematics
 Curry–Howard correspondence
Editors and Affiliations

Institute of Philosophy, Technical University of Berlin, Berlin, Germany
Stefania Centrone

Department of Philosophy, University of Helsinki, Helsinki, Finland
Sara Negri

University of Hamburg, Hamburg, Germany
Deniz Sarikaya

Dipartimento di Informatica, Università degli Studi di Verona, Verona, Italy
Peter M. Schuster
About the editors
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.
Bibliographic Information
Book Title: Mathesis Universalis, Computability and Proof
Editors: Stefania Centrone, Sara Negri, Deniz Sarikaya, Peter M. Schuster
Series Title: Synthese Library
DOI: https://doi.org/10.1007/9783030204471
Publisher: Springer Cham
eBook Packages: Religion and Philosophy, Philosophy and Religion (R0)
Copyright Information: Springer Nature Switzerland AG 2019
Hardcover ISBN: 9783030204464
Softcover ISBN: 9783030204495
eBook ISBN: 9783030204471
Series ISSN: 01666991
Series EISSN: 25428292
Edition Number: 1
Number of Pages: X, 374
Number of Illustrations: 38 b/w illustrations
Topics: Logic, Mathematical Logic and Foundations, Formal Languages and Automata Theory