Mathesis Universalis, Computability and Proof

• Συγγραφή απο Οργανισμό/Αρχή: SpringerLink (Online service)
• Άλλοι συγγραφείς: Centrone, Stefania (Επιμελητής έκδοσης, http://id.loc.gov/vocabulary/relators/edt), Negri, Sara (Επιμελητής έκδοσης, http://id.loc.gov/vocabulary/relators/edt), Sarikaya, Deniz (Επιμελητής έκδοσης, http://id.loc.gov/vocabulary/relators/edt), Schuster, Peter M. (Επιμελητής έκδοσης, http://id.loc.gov/vocabulary/relators/edt)
• Μορφή: Ηλεκτρονική πηγή Ηλ. βιβλίο
• Γλώσσα: English
• Έκδοση: Cham : Springer International Publishing : Imprint: Springer, 2019.
• Έκδοση: 1st ed. 2019.
• Σειρά: Synthese Library, Studies in Epistemology, Logic, Methodology, and Philosophy of Science ; 412
• Θέματα: Logic.; Mathematical logic.; Mathematical Logic and Foundations.; Mathematical Logic and Formal Languages.

 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. 

 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 λ-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). .