Mathesis Universalis, Computability and Proof
Springer International Publishing (Verlag)
978-3-030-20446-4 (ISBN)
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. 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, intuitionisticlogic, constructivism and reverse mathematics, delving deeply into a contextual examination of the relationship between mathematical rigor and demands for simplification.
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).
1. Introduction: Mathesis Universalis, Proof and Computation (Stefania Centrone).- 2. Diplomacy of Trust in the European Crisis (Enno Aufderheide).- 3. Mathesis Universalis and Homotopy Type Theory (Steve Awodey).- 4. Note on the Benefit of Proof Representations by Name (Matthias Baaz).- 5. Constructive Proofs of Negated Statements (Josef Berger and Gregor Svindland).- 6. Constructivism in Abstract Mathematics (Ulrich Berger).- 7. Addressing Circular Definitions via Systems of Proofs (Riccardo Bruni).- 8. The Monotone Completeness Theorem in Constructive Reverse Mathematics (Hajime Ishihara and Takako Nemoto).- 9. From Mathesis Universalis to Fixed Points and Related Set-Theoretic Concepts (Gerhard Jäger and Silvia Steila).- 10. Through an Inference Rule, Darkly (Roman Kuznets).- 11. Objectivity and Truth in Mathematics: A Sober Non-Platonist Perspective (Godehard Link).- 12. From Mathesis Universalis to Provability, Computability, and Constructivity (Klaus Mainzer).- 13. Analytic Equational Proof Systems for Combinatory Logic and -Calculus: a Survey (Pierluigi Minari).- 14. Computational Interpretations of Classical Reasoning: From the Epsilon Calculus to Stateful Programs (Thomas Powell).- 15. The Concepts of Proof and Ground (Dag Prawitz).- 16. On Relating Theories: Proof-Theoretical Reduction (Michael Rathjen and Michael Toppel).- 17. Program Extraction from Proofs: the Fan Theorem for Uniformly Coconvex Bars (Helmut Schwichtenberg).- 18. Counting and Numbers, from Pure Mathesis to Base Conversion Algorithms (Jan von Plato).- 19. Point-Free Spectra of Linear Spreads (Daniel Wessel).
| Erscheinungsdatum | 08.11.2019 |
|---|---|
| Reihe/Serie | Synthese Library |
| Zusatzinfo | X, 374 p. 38 illus. |
| Verlagsort | Cham |
| Sprache | englisch |
| Maße | 155 x 235 mm |
| Gewicht | 733 g |
| Themenwelt | Geisteswissenschaften ► Philosophie ► Allgemeines / Lexika |
| Geisteswissenschaften ► Philosophie ► Logik | |
| Mathematik / Informatik ► Informatik ► Theorie / Studium | |
| Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre | |
| Schlagworte | Analytic Philosophy mathematics • Bolzano philosophy • Calculus Ratiocinator • Characteristica universalis • Concept of Mathematics and Classification • Constructive Mathematics • Craig's interpolation theorem • Curry-Howard correspondence • Curry–Howard correspondence • Foundations of mathematics • Gottfried Wilhelm Leibniz • History of Mathematics philosophy • intensional type theory • Mathesis • Mathesis Universalis • metamathematics • ordinal analysis • philosophy of mathematics • Proof Theory logic • reverse mathematics • Turing Machine Philosophy |
| ISBN-10 | 3-030-20446-4 / 3030204464 |
| ISBN-13 | 978-3-030-20446-4 / 9783030204464 |
| Zustand | Neuware |
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