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This book provides an overview of the confluence of ideas in
Turing's era and work and examines the impact of his work on
mathematical logic and theoretical computer science. It combines
contributions by well-known scientists on the history and
philosophy of computability theory as well as on generalised Turing
computability. By looking at the roots and at the philosophical and
technical influence of Turing's work, it is possible to gather new
perspectives and new research topics which might be considered as a
continuation of Turing's working ideas well into the 21st century.
The Stored-Program Universal Computer: Did Zuse Anticipate Turing
and von Neumann?" is available open access under a Creative Commons
Attribution 4.0 International License via link.springer.com
A comprehensive survey of Martin-Loef's constructive type theory,
considerable parts of which have only been presented by Martin-Loef
in lecture form or as part of conference talks. Sommaruga surveys
the prehistory of type theory and its highly complex development
through eight different stages from 1970 to 1995. He also provides
a systematic presentation of the latest version of the theory, as
offered by Martin-Loef at Leiden University in Fall 1993. This
presentation gives a fuller and updated account of the system.
Earlier, brief presentations took no account of the issues related
to the type-theoretical approach to logic and the foundations of
mathematics, while here they are accorded an entire part of the
book. Readership: Comprehensive accounts of the history and
philosophy of constructive type theory and a considerable amount of
related material. Readers need a solid background in standard logic
and a first, basic acquaintance with type theory.
The book "Foundational Theories of Classical and Constructive
Mathematics" is a book on the classical topic of foundations of
mathematics. Its originality resides mainly in its treating at the
same time foundations of classical and foundations of constructive
mathematics. This confrontation of two kinds of foundations
contributes to answering questions such as: Are
foundations/foundational theories of classical mathematics of a
different nature compared to those of constructive mathematics? Do
they play the same role for the resp. mathematics? Are there
connections between the two kinds of foundational theories? etc.
The confrontation and comparison is often implicit and sometimes
explicit. Its great advantage is to extend the traditional
discussion of the foundations of mathematics and to render it at
the same time more subtle and more differentiated. Another
important aspect of the book is that some of its contributions are
of a more philosophical, others of a more technical nature. This
double face is emphasized, since foundations of mathematics is an
eminent topic in the philosophy of mathematics: hence both sides of
this discipline ought to be and are being paid due to.
In this two-volume compilation of articles, leading researchers
reevaluate the success of Hilbert's axiomatic method, which not
only laid the foundations for our understanding of modern
mathematics, but also found applications in physics, computer
science and elsewhere. The title takes its name from David
Hilbert's seminal talk Axiomatisches Denken, given at a meeting of
the Swiss Mathematical Society in Zurich in 1917. This marked the
beginning of Hilbert's return to his foundational studies, which
ultimately resulted in the establishment of proof theory as a new
branch in the emerging field of mathematical logic. Hilbert also
used the opportunity to bring Paul Bernays back to Goettingen as
his main collaborator in foundational studies in the years to come.
The contributions are addressed to mathematical and philosophical
logicians, but also to philosophers of science as well as
physicists and computer scientists with an interest in foundations.
In this two-volume compilation of articles, leading researchers
reevaluate the success of Hilbert's axiomatic method, which not
only laid the foundations for our understanding of modern
mathematics, but also found applications in physics, computer
science and elsewhere. The title takes its name from David
Hilbert's seminal talk Axiomatisches Denken, given at a meeting of
the Swiss Mathematical Society in Zurich in 1917. This marked the
beginning of Hilbert's return to his foundational studies, which
ultimately resulted in the establishment of proof theory as a new
branch in the emerging field of mathematical logic. Hilbert also
used the opportunity to bring Paul Bernays back to Goettingen as
his main collaborator in foundational studies in the years to come.
The contributions are addressed to mathematical and philosophical
logicians, but also to philosophers of science as well as
physicists and computer scientists with an interest in foundations.
Chapter 8 is available open access under a Creative Commons
Attribution 4.0 International License via link.springer.com.
This book provides an overview of the confluence of ideas in
Turing's era and work and examines the impact of his work on
mathematical logic and theoretical computer science. It combines
contributions by well-known scientists on the history and
philosophy of computability theory as well as on generalised Turing
computability. By looking at the roots and at the philosophical and
technical influence of Turing's work, it is possible to gather new
perspectives and new research topics which might be considered as a
continuation of Turing's working ideas well into the 21st century.
The Stored-Program Universal Computer: Did Zuse Anticipate Turing
and von Neumann?" is available open access under a Creative Commons
Attribution 4.0 International License via link.springer.com
The book "Foundational Theories of Classical and Constructive
Mathematics" is a book on the classical topic of foundations of
mathematics. Its originality resides mainly in its treating at the
same time foundations of classical and foundations of constructive
mathematics. This confrontation of two kinds of foundations
contributes to answering questions such as: Are
foundations/foundational theories of classical mathematics of a
different nature compared to those of constructive mathematics? Do
they play the same role for the resp. mathematics? Are there
connections between the two kinds of foundational theories? etc.
The confrontation and comparison is often implicit and sometimes
explicit. Its great advantage is to extend the traditional
discussion of the foundations of mathematics and to render it at
the same time more subtle and more differentiated. Another
important aspect of the book is that some of its contributions are
of a more philosophical, others of a more technical nature. This
double face is emphasized, since foundations of mathematics is an
eminent topic in the philosophy of mathematics: hence both sides of
this discipline ought to be and are being paid due to.
A comprehensive survey of Martin-Loef's constructive type theory,
considerable parts of which have only been presented by Martin-Loef
in lecture form or as part of conference talks. Sommaruga surveys
the prehistory of type theory and its highly complex development
through eight different stages from 1970 to 1995. He also provides
a systematic presentation of the latest version of the theory, as
offered by Martin-Loef at Leiden University in Fall 1993. This
presentation gives a fuller and updated account of the system.
Earlier, brief presentations took no account of the issues related
to the type-theoretical approach to logic and the foundations of
mathematics, while here they are accorded an entire part of the
book. Readership: Comprehensive accounts of the history and
philosophy of constructive type theory and a considerable amount of
related material. Readers need a solid background in standard logic
and a first, basic acquaintance with type theory.
It is commonly assumed that computers process information. But what
is inf- mation? In a technical, important, but nevertheless rather
narrow sense, Sh-
non'sinformationtheorygivesa?rstanswertothisquestion.Thistheoryfocuses
on measuring the information content of a message. Essentially this
measure is the reduction of the uncertainty obtained by receiving a
message. The unc- tainty of a situation of ignorance in turn is
measured by entropy. This theory hashad an immense impact on the
technologyof information storage,data c- pression, information
transmission and coding and still is a very active domain of
research. Shannon's theory has also attractedmuch interest in a
more philosophic look at information, although it was readily
remarked that it is only a "syntactic" theory of information and
neglects "semantic" issues. Several attempts have been made in
philosophy to give information theory a semantic ?avor, but still
mostly based on or at least linked to Shannon's theory. Approaches
to semantic
informationtheoryalsoveryoftenmakeuseofformallogic.Thereby,information
is linked to reasoning, deduction and inference, as well as to
decision making. Further, entropy and related measure were soon
found to have important connotations with regard to statistical
inference. Surely, statistical data and observation represent
information, information about unknown, hidden para- ters. Thus a
whole branch of statistics developed around concepts of Shannon's
information theory or derived from them. Also some proper
measurements - propriate for statistics, like Fisher's information,
were proposed.
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