LEADER 05919nam 22005655 450 001 9910349539003321 005 20200813130747.0 010 $a3-030-20447-2 024 7 $a10.1007/978-3-030-20447-1 035 $a(CKB)4100000009670340 035 $a(MiAaPQ)EBC5966954 035 $a(DE-He213)978-3-030-20447-1 035 $a(EXLCZ)994100000009670340 100 $a20191025d2019 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aMathesis Universalis, Computability and Proof /$fedited by Stefania Centrone, Sara Negri, Deniz Sarikaya, Peter M. Schuster 205 $a1st ed. 2019. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2019. 215 $a1 online resource (375 pages) 225 1 $aSynthese Library, Studies in Epistemology, Logic, Methodology, and Philosophy of Science,$x0166-6991 ;$v412 311 $a3-030-20446-4 327 $a1. 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). . 330 $aIn a fragment entitled Elementa Nova Matheseos Universalis (1683?) Leibniz writes ?the mathesis [?] shall deliver the method through which things that are conceivable can be exactly determined?; in another fragment he takes the mathesis to be ?the science of all things that are conceivable.? Leibniz considers all mathematical disciplines as branches of the mathesis and conceives the mathesis as a general science of forms applicable not only to magnitudes but to every object that exists in our imagination, i.e. that is possible at least in principle. As a general science of forms the mathesis investigates possible relations between ?arbitrary objects? (?objets quelconques?). It is an abstract theory of combinations and relations among objects whatsoever. In 1810 the mathematician and philosopher Bernard Bolzano published a booklet entitled Contributions to a Better-Grounded Presentation of Mathematics. There is, according to him, a certain objective connection among the truths that are germane to a certain homogeneous field of objects: some truths are the ?reasons? (?Gründe?) of others, and the latter are ?consequences? (?Folgen?) of the former. The reason-consequence relation seems to be the counterpart of causality at the level of a relation between true propositions. A rigorous proof is characterized in this context as a proof that shows the reason of the proposition that is to be proven. Requirements imposed on rigorous proofs seem to anticipate normalization results in current proof theory. The contributors of Mathesis Universalis, Computability and Proof, leading experts in the fields of computer science, mathematics, logic and philosophy, show the evolution of these and related ideas exploring topics in proof theory, computability theory, intuitionistic logic, constructivism and reverse mathematics, delving deeply into a contextual examination of the relationship between mathematical rigor and demands for simplification. 410 0$aSynthese Library, Studies in Epistemology, Logic, Methodology, and Philosophy of Science,$x0166-6991 ;$v412 606 $aLogic 606 $aMathematical logic 606 $aLogic$3https://scigraph.springernature.com/ontologies/product-market-codes/E16000 606 $aMathematical Logic and Foundations$3https://scigraph.springernature.com/ontologies/product-market-codes/M24005 606 $aMathematical Logic and Formal Languages$3https://scigraph.springernature.com/ontologies/product-market-codes/I16048 615 0$aLogic. 615 0$aMathematical logic. 615 14$aLogic. 615 24$aMathematical Logic and Foundations. 615 24$aMathematical Logic and Formal Languages. 676 $a511.3 702 $aCentrone$b Stefania$4edt$4http://id.loc.gov/vocabulary/relators/edt 702 $aNegri$b Sara$4edt$4http://id.loc.gov/vocabulary/relators/edt 702 $aSarikaya$b Deniz$4edt$4http://id.loc.gov/vocabulary/relators/edt 702 $aSchuster$b Peter M$4edt$4http://id.loc.gov/vocabulary/relators/edt 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910349539003321 996 $aMathesis Universalis, Computability and Proof$92102279 997 $aUNINA