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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 edited work presents contemporary mathematical practice in the
foundational mathematical theories, in particular set theory and
the univalent foundations. It shares the work of significant
scholars across the disciplines of mathematics, philosophy and
computer science. Readers will discover systematic thought on
criteria for a suitable foundation in mathematics and philosophical
reflections around the mathematical perspectives. The volume is
divided into three sections, the first two of which focus on the
two most prominent candidate theories for a foundation of
mathematics. Readers may trace current research in set theory,
which has widely been assumed to serve as a framework for
foundational issues, as well as new material elaborating on the
univalent foundations, considering an approach based on homotopy
type theory (HoTT). The third section then builds on this and is
centred on philosophical questions connected to the foundations of
mathematics. Here, the authors contribute to discussions on
foundational criteria with more general thoughts on the foundations
of mathematics which are not connected to particular theories. This
book shares the work of some of the most important scholars in the
fields of set theory (S. Friedman), non-classical logic (G. Priest)
and the philosophy of mathematics (P. Maddy). The reader will
become aware of the advantages of each theory and objections to it
as a foundation, following the latest and best work across the
disciplines and it is therefore a valuable read for anyone working
on the foundations of mathematics or in the philosophy of
mathematics.
Mathematics and mathematics education research have an ongoing
interest in improving our understanding of mathematical problem
posing and solving. This book focuses on problem posing in a
context of mathematical giftedness. The contributions particularly
address where such problems come from, what properties they should
have, and which differences between school mathematics and more
complex kinds of mathematics exist. These perspectives are examined
internationally, allowing for cross-national insights.
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 edited work presents contemporary mathematical practice in the
foundational mathematical theories, in particular set theory and
the univalent foundations. It shares the work of significant
scholars across the disciplines of mathematics, philosophy and
computer science. Readers will discover systematic thought on
criteria for a suitable foundation in mathematics and philosophical
reflections around the mathematical perspectives. The volume is
divided into three sections, the first two of which focus on the
two most prominent candidate theories for a foundation of
mathematics. Readers may trace current research in set theory,
which has widely been assumed to serve as a framework for
foundational issues, as well as new material elaborating on the
univalent foundations, considering an approach based on homotopy
type theory (HoTT). The third section then builds on this and is
centred on philosophical questions connected to the foundations of
mathematics. Here, the authors contribute to discussions on
foundational criteria with more general thoughts on the foundations
of mathematics which are not connected to particular theories. This
book shares the work of some of the most important scholars in the
fields of set theory (S. Friedman), non-classical logic (G. Priest)
and the philosophy of mathematics (P. Maddy). The reader will
become aware of the advantages of each theory and objections to it
as a foundation, following the latest and best work across the
disciplines and it is therefore a valuable read for anyone working
on the foundations of mathematics or in the philosophy of
mathematics.
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