Centrone S., Negri S., Sarikaya D., Schuster P.M. (Eds.) Mathesis Universalis, Computability and Proof

Springer, 2019. — 375 p. — (Synthese Library: Studies in Epistemology, Logic, Methodology, and Philosophy of Science 412). — ISBN: 978-3-030-20446-4.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.Introduction: Mathesis Universalis, Proof and Computation
Diplomacy of Trust in the European Crisis: Contributions by the Alexander von Humboldt Foundation
Mathesis Universalis and Homotopy Type Theory
Note on the Benefit of Proof Representations by Name
Constructive Proofs of Negated Statements
On the Constructive and Computational Content of Abstract Mathematics
Addressing Circular Definitions via Systems of Proofs
The Monotone Completeness Theorem in Constructive Reverse Mathematics
From Mathesis Universalis to Fixed Points and Related Set-Theoretic Concepts
Through an Inference Rule, Darkly
Objectivity and Truth in Mathematics: A Sober Non-platonist Perspective
From Mathesis Universalis to Provability, Computability, and Constructivity
Analytic Equational Proof Systems for Combinatory Logic and λ-Calculus:A Survey
Computational Interpretations of Classical Reasoning: From the Epsilon Calculus to Stateful Programs
The Concepts of Proof and Ground
On Relating Theories: Proof-Theoretical Reduction
Program Extraction from Proofs: The Fan Theorem for Uniformly Coconvex Bars
Counting and Numbers, from Pure Mathesis to Base Conversion Algorithms
Point-Free Spectra of Linear Spreads