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, intuitionisticlogic, constructivism and reverse mathematics, delving deeply into a contextual examination of the relationship between mathematical rigor and demands for simplification.

Stefania Centrone is currently Privatdozentin at the University of Hamburg, teaches and does research at the Universities of Oldenburg and of Helsinki and has been in 2016 deputy Professor of Theoretical Philosophy at the University of Göttingen. In 2012 she was awarded a DFG-Eigene Stelle for the project Bolzanos und Husserls Weiterentwicklung von Leibnizens Ideen zur Mathesis Universalis and 2017 a Heisenberg grant. She is author of the volumes Logic and philosophy of Mathematics in the Early Husserl (Synthese Library 2010) and Studien zu Bolzano (Academia Verlag 2015).

Sara Negri is Professor of Theoretical Philosophy at the University of Helsinki, where she has been a Docent of Logic since 1998. After a PhD in Mathematics in 1996 at the University of Padova and research visits at the University of Amsterdam and Chalmers, she has been a research associate at the Imperial College in London, a Humboldt Fellow in Munich, and a visiting scientist at the Mittag-Leffler Institute in Stockholm. Her research interests range from mathematical logic and philosophy of mathematics to proof theory and its applications to philosophical logic and formal epistemology.

Deniz Sarikaya is PhD-Student of Philosophy and studies Mathematics at the University of Hamburg with experience abroad at the Universiteit van Amsterdam and Universidad de Barcelona. He stayed a term as a Visiting Student Researcher at the University of California, Berkeley developing a project on the Philosophy of Mathematical Practice concerning the Philosophical impact of the usage of automatic theorem prover and as a RISE research intern at the University of British Columbia. He is mainly focusing on philosophy of mathematics and logic.

Peter Schuster is Associate Professor for Mathematical Logic at the University of Verona. After both doctorate and habilitation in mathematics at the University of Munich he was Lecturer at the University of Leeds and member of the Leeds Logic Group. Apart from constructive mathematics at large, his principal research interests are about the computational content of classical proofs in abstract algebra and related fields in which maximum or minimum principles are invoked.

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 ¿-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).