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Mathesis Universalis, Computability and Proof

  • Gottfried Leibniz's philosophy of logic in the Digital Age

  • Views on second-order 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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  • ISBN: 978-3-030-20447-1
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Table of contents (19 chapters)

  1. Front Matter

    Pages i-x
  2. Constructive Proofs of Negated Statements

    • Josef Berger, Gregor Svindland
    Pages 47-53
  3. The Monotone Completeness Theorem in Constructive Reverse Mathematics

    • Hajime Ishihara, Takako Nemoto
    Pages 101-112
  4. Through an Inference Rule, Darkly

    • Roman Kuznets
    Pages 131-158
  5. The Concepts of Proof and Ground

    • Dag Prawitz
    Pages 291-309
  6. On Relating Theories: Proof-Theoretical Reduction

    • Michael Rathjen, Michael Toppel
    Pages 311-331
  7. Point-Free Spectra of Linear Spreads

    • Daniel Wessel
    Pages 353-374

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 Better-Grounded 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 reason-consequence 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

Reviews

   

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 DFG-Eigene 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 Mittag-Leffler 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 PhD-Student 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.

Bibliographic Information

Buying options

eBook USD 109.00
Price excludes VAT (USA)
  • ISBN: 978-3-030-20447-1
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 149.99
Price excludes VAT (USA)
Hardcover Book USD 149.99
Price excludes VAT (USA)