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In the four decades since Imre Lakatos declared mathematics a
"quasi-empirical science," increasing attention has been paid to
the process of proof and argumentation in the field -- a
development paralleled by the rise of computer technology and the
mounting interest in the logical underpinnings of mathematics.
Explanantion and Proof in Mathematics assembles perspectives from
mathematics education and from the philosophy and history of
mathematics to strengthen mutual awareness and share recent
findings and advances in their interrelated fields. With examples
ranging from the geometrists of the 17th century and ancient
Chinese algorithms to cognitive psychology and current educational
practice, contributors explore the role of refutation in generating
proofs, the varied links between experiment and deduction, the use
of diagrammatic thinking in addition to pure logic, and the uses of
proof in mathematics education (including a critique of
"authoritative" versus "authoritarian" teaching styles). A sampling
of the coverage: The conjoint origins of proof and theoretical
physics in ancient Greece. Proof as bearers of mathematical
knowledge. Bridging knowing and proving in mathematical reasoning.
The role of mathematics in long-term cognitive development of
reasoning. Proof as experiment in the work of Wittgenstein.
Relationships between mathematical proof, problem-solving, and
explanation. Explanation and Proof in Mathematics is certain to
attract a wide range of readers, including mathematicians,
mathematics education professionals, researchers, students, and
philosophers and historians of mathematics.
This open access book shares revealing insights into the
development of mathematics education research in Germany from 1976
(ICME 3 in Karlsruhe) to 2016 (ICME 13 in Hamburg). How did
mathematics education research evolve in the course of these four
decades? Which ideas and people were most influential, and how did
German research interact with the international community? These
questions are answered by scholars from a range of fields and in
ten thematic sections: (1) a short survey of the development of
educational research on mathematics in German speaking countries
(2) subject-matter didactics, (3) design science and design
research, (4) modelling, (5) mathematics and Bildung 1810 to 1850,
(6) Allgemeinbildung, Mathematical Literacy, and Competence
Orientation (7) theory traditions, (8) classroom studies, (9)
educational research and (10) large-scale studies. During the time
span presented here, profound changes took place in German-speaking
mathematics education research. Besides the traditional fields of
activity like subject-matter didactics or design science,
completely new areas also emerged, which are characterized by
various empirical approaches and a closer connection to psychology,
sociology, epistemology and general education research. Each
chapter presents a respective area of mathematics education in
Germany and analyzes its relevance for the development of the
research community, not only with regard to research findings and
methods but also in terms of interaction with the educational
system. One of the central aspects in all chapters concerns the
constant efforts to find common ground between mathematics and
education. In addition, readers can benefit from this analysis by
comparing the development shown here with the mathematical
education research situation in their own country.
In the four decades since Imre Lakatos declared mathematics a
"quasi-empirical science," increasing attention has been paid to
the process of proof and argumentation in the field -- a
development paralleled by the rise of computer technology and the
mounting interest in the logical underpinnings of mathematics.
Explanantion and Proof in Mathematics assembles perspectives from
mathematics education and from the philosophy and history of
mathematics to strengthen mutual awareness and share recent
findings and advances in their interrelated fields. With examples
ranging from the geometrists of the 17th century and ancient
Chinese algorithms to cognitive psychology and current educational
practice, contributors explore the role of refutation in generating
proofs, the varied links between experiment and deduction, the use
of diagrammatic thinking in addition to pure logic, and the uses of
proof in mathematics education (including a critique of
"authoritative" versus "authoritarian" teaching styles).
A sampling of the coverage:
- The conjoint origins of proof and theoretical physics in
ancient Greece.
- Proof as bearers of mathematical knowledge.
- Bridging knowing and proving in mathematical reasoning.
- The role of mathematics in long-term cognitive development of
reasoning.
- Proof as experiment in the work of Wittgenstein.
- Relationships between mathematical proof, problem-solving, and
explanation.
Explanation and Proof in Mathematics is certain to attract a
wide range of readers, including mathematicians, mathematics
education professionals, researchers, students, and philosophers
and historians of mathematics.
"Geschichte der Analysis" ist von einem internationalen
Expertenteam geschrieben und stellt die gegenwartig umfassendste
Darstellung der Herausbildung und Entwicklung dieser mathematischen
Kerndisziplin dar. Der tiefgreifende begriffliche Wandel, den die
Analysis im Laufe der Zeit durchgemacht hat, wird ebenso
dargestellt, wie auch der Einfluss, den vor allem physikalische
Probleme gehabt haben. Biographische und philosophische
Hintergrunde werden ausgeleuchtet und ihre Relevanz fur die
Theorieentwicklung gezeigt. Neben der eigentlichen Geschichte der
Analysis bis ungefahr 1900 enthalt das Buch Spezialkapitel uber die
Entwicklung der analytischen Mechanik im 18. Jahrhundert,
Randwertprobleme der mathematischen Physik im 19. Jahrhundert, die
Theorie der komplexen Funktionen, die Grundlagenkrise sowie
historische Uberblicke uber die Variationsrechnung,
Differentialgleichungen und Funktionalanalysis."
Analysis as an independent subject was created as part of the
scientific revolution in the seventeenth century. Kepler, Galileo,
Descartes, Fermat, Huygens, Newton, and Leibniz, to name but a few,
contributed to its genesis. Since the end of the seventeenth
century, the historical progress of mathematical analysis has
displayed unique vitality and momentum. No other mathematical field
has so profoundly influenced the development of modern scientific
thinking. Describing this multidimensional historical development
requires an in-depth discussion which includes a reconstruction of
general trends and an examination of the specific problems.This
volume is designed as a collective work of authors who are proven
experts in the history of mathematics. It clarifies the conceptual
change that analysis underwent during its development while
elucidating the influence of specific applications and describing
the relevance of biographical and philosophical backgrounds. The
first ten chapters of the book outline chronological development
and the last three chapters survey the history of differential
equations, the calculus of variations, and functional analysis.
Special features are a separate chapter on the development of the
theory of complex functions in the nineteenth century and two
chapters on the influence of physics on analysis.One is about the
origins of analytical mechanics, and one treats the development of
boundary-value problems of mathematical physics (especially
potential theory) in the nineteenth century. The book presents an
accurate and very readable account of the history of analysis. Each
chapter provides a comprehensive bibliography. Mathematical
examples have been carefully chosen so that readers with a modest
background in mathematics can follow them. It is suitable for
mathematical historians and a general mathematical audience.
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