Mathesis universalis, computability and proof
著者
書誌事項
Mathesis universalis, computability and proof
（Synthese library, v. 412）
Springer, c2019
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注記
Includes bibliographical references
内容説明・目次
内容説明
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" ("Grunde") 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.
目次
1. Introduction: Mathesis Universalis, Proof and ComputationStefania Centrone
2. Diplomacy of Trust in the European CrisisEnno Aufderheide
3. Mathesis Universalis and Homotopy Type TheorySteve Awodey
4. Note on the Benefit of Proof Representations by NameMatthias Baaz
5. Constructive Proofs of Negated StatementsJosef Berger and Gregor Svindland
6. Constructivism in Abstract MathematicsUlrich Berger
7. Addressing Circular Definitions via Systems of ProofsRiccardo Bruni
8. The Monotone Completeness Theorem in Constructive Reverse MathematicsHajime Ishihara and Takako Nemoto
9. From Mathesis Universalis to Fixed Points and Related Set-Theoretic ConceptsGerhard Jager and Silvia Steila
10. Through an Inference Rule, DarklyRoman Kuznets
11. Objectivity and Truth in Mathematics: A Sober Non-Platonist PerspectiveGodehard Link
12. From Mathesis Universalis to Provability, Computability, and ConstructivityKlaus Mainzer
13. Analytic Equational Proof Systems for Combinatory Logic and -Calculus: a SurveyPierluigi Minari
14. Computational Interpretations of Classical Reasoning: From the Epsilon Calculus to Stateful ProgramsThomas Powell
15. The Concepts of Proof and GroundDag Prawitz
16. On Relating Theories: Proof-Theoretical ReductionMichael Rathjen and Michael Toppel
17. Program Extraction from Proofs: the Fan Theorem for Uniformly Coconvex BarsHelmut Schwichtenberg
18. Counting and Numbers, from Pure Mathesis to Base Conversion AlgorithmsJan von Plato
19. Point-Free Spectra of Linear SpreadsDaniel Wessel
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