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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.
This book continues from where the authors' previous book,
Structural Proof Theory, ended. It presents an extension of the
methods of analysis of proofs in pure logic to elementary axiomatic
systems and to what is known as philosophical logic. A
self-contained brief introduction to the proof theory of pure logic
is included that serves both the mathematically and philosophically
oriented reader. The method is built up gradually, with examples
drawn from theories of order, lattice theory and elementary
geometry. The aim is, in each of the examples, to help the reader
grasp the combinatorial behaviour of an axiom system, which
typically leads to decidability results. The last part presents, as
an application and extension of all that precedes it, a
proof-theoretical approach to the Kripke semantics of modal and
related logics, with a great number of new results, providing
essential reading for mathematical and philosophical logicians.
Structural proof theory is a branch of logic that studies the general structure and properties of logical and mathematical proofs. This book is both a concise introduction to the central results and methods of structural proof theory, and a work of research that will be of interest to specialists. The book is designed to be used by students of philosophy, mathematics, and computer science. A special feature of the volume is a computerized system for developing proofs interactively, downloadable from the web and regularly updated.
This book continues from where the authors' previous book,
Structural Proof Theory, ended. It presents an extension of the
methods of analysis of proofs in pure logic to elementary axiomatic
systems and to what is known as philosophical logic. A
self-contained brief introduction to the proof theory of pure logic
is included that serves both the mathematically and philosophically
oriented reader. The method is built up gradually, with examples
drawn from theories of order, lattice theory and elementary
geometry. The aim is, in each of the examples, to help the reader
grasp the combinatorial behaviour of an axiom system, which
typically leads to decidability results. The last part presents, as
an application and extension of all that precedes it, a
proof-theoretical approach to the Kripke semantics of modal and
related logics, with a great number of new results, providing
essential reading for mathematical and philosophical logicians.
Structural proof theory is a branch of logic that studies the
general structure and properties of logical and mathematical
proofs. This book is both a concise introduction to the central
results and methods of structural proof theory, and a work of
research that will be of interest to specialists. The book is
designed to be used by students of philosophy, mathematics and
computer science. The book contains a wealth of results on
proof-theoretical systems, including extensions of such systems
from logic to mathematics, and on the connection between the two
main forms of structural proof theory - natural deduction and
sequent calculus. The authors emphasize the computational content
of logical results. A special feature of the volume is a
computerized system for developing proofs interactively,
downloadable from the web and regularly updated.
This book constitutes the proceedings of the 30th International
Conference on Automated Reasoning with Analytic Tableaux and
Related Methods, TABLEAUX 2021, held in Birmingham, UK, in
September 2021.The 23 full papers and 3 system descriptions
included in the volume were carefully reviewed and selected from 46
submissions.They present research on all aspects of the
mechanization of tableaux-based reasoning and related methods,
including theoretical foundations, implementation techniques,
systems development and applications. The papers are organized in
the following topical sections: tableau calculi, sequent calculi,
theorem proving, formalized proofs, non-wellfounded proofs,
automated theorem provers, and intuitionistic modal logics.
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.
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