Mathesis Universalis, Computability and Proof

Description

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, intuitionistic logic, constructivism and reverse mathematics, delving deeply into a contextual examination of the relationship between mathematical rigor and demands for simplification. Front Matter ....Pages i-x Introduction: Mathesis Universalis, Proof and Computation (Stefania Centrone)....Pages 1-6 Diplomacy of Trust in the European Crisis: Contributions by the Alexander von Humboldt Foundation (Enno Aufderheide)....Pages 7-11 Mathesis Universalis and Homotopy Type Theory (Steve Awodey)....Pages 13-36 Note on the Benefit of Proof Representations by Name (Matthias Baaz)....Pages 37-45 Constructive Proofs of Negated Statements (Josef Berger, Gregor Svindland)....Pages 47-53 On the Constructive and Computational Content of Abstract Mathematics (Ulrich Berger)....Pages 55-73 Addressing Circular Definitions via Systems of Proofs (Riccardo Bruni)....Pages 75-100 The Monotone Completeness Theorem in Constructive Reverse Mathematics (Hajime Ishihara, Takako Nemoto)....Pages 101-112 From Mathesis Universalis to Fixed Points and Related Set-Theoretic Concepts (Gerhard Jäger, Silvia Steila)....Pages 113-129 Through an Inference Rule, Darkly (Roman Kuznets)....Pages 131-158 Objectivity and Truth in Mathematics: A Sober Non-platonist Perspective (Godehard Link)....Pages 159-202 From Mathesis Universalis to Provability, Computability, and Constructivity (Klaus Mainzer)....Pages 203-234 Analytic Equational Proof Systems for Combinatory Logic and λ-Calculus:A Survey (Pierluigi Minari)....Pages 235-253 Computational Interpretations of Classical Reasoning: From the Epsilon Calculus to Stateful Programs (Thomas Powell)....Pages 255-290 The Concepts of Proof and Ground (Dag Prawitz)....Pages 291-309 On Relating Theories: Proof-Theoretical Reduction (Michael Rathjen, Michael Toppel)....Pages 311-331 Program Extraction from Proofs: The Fan Theorem for Uniformly Coconvex Bars (Helmut Schwichtenberg)....Pages 333-341 Counting and Numbers, from Pure Mathesis to Base Conversion Algorithms (Jan von Plato)....Pages 343-351 Point-Free Spectra of Linear Spreads (Daniel Wessel)....Pages 353-374