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Books > Science & Mathematics > Mathematics > Philosophy of mathematics
Does 2 + 2 = 4? Ask almost anyone and the answer will be an
unequivocal yes. A basic equation such as this seems the very
definition of certainty, but how is this so?
In this captivating book, Helen Verran addresses precisely that
question by looking at how science, mathematics, and logic come to
life in Yoruba primary schools. Drawing on her experience as a
teacher in Nigeria, Verran describes how she went from the radical
conclusion that logic and math are culturally relative, to
determining what Westerners find so disconcerting about Yoruba
logic and to a new understanding of all generalizing logic. She
reveals that in contrast to the one-to-many model found in Western
number systems, Yoruba thinking operates by figuring things as
wholes and their parts. Quantity is not absolute but always
relational. Certainty derives not from abstract logic, but from
cultural practice and association.
A powerful story of how one woman's investigation into an everyday
African situation led to extraordinary conclusions about the nature
of numbers, generalization, and certainty, this book will be a
signal contribution to philosophy, anthropology of science, and
education.
Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer
Book Archives mit Publikationen, die seit den Anfangen des Verlags
von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv
Quellen fur die historische wie auch die disziplingeschichtliche
Forschung zur Verfugung, die jeweils im historischen Kontext
betrachtet werden mussen. Dieser Titel erschien in der Zeit vor
1945 und wird daher in seiner zeittypischen politisch-ideologischen
Ausrichtung vom Verlag nicht beworben.
Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer
Book Archives mit Publikationen, die seit den Anfangen des Verlags
von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv
Quellen fur die historische wie auch die disziplingeschichtliche
Forschung zur Verfugung, die jeweils im historischen Kontext
betrachtet werden mussen. Dieser Titel erschien in der Zeit vor
1945 und wird daher in seiner zeittypischen politisch-ideologischen
Ausrichtung vom Verlag nicht beworben.
Dieses Bandchen setzt in keiner Weise mathematische Kenntnisse
voraus und mOchte sich an jeden wenden, der mit Verstandnis rechnen
gelernt und die Preude daran nicht verloren hat. Seiner ganzen
Passung nach durfte es viel leicht den Liebhabern der
Kulturgeschichte nicht unwill kommen sein. Dasseldorf, Weihnachten
1922. Ewald Fettwel8. INHALTSVERZEICHNIS 1. Oie ersten Anfange des
Zahlens und Rechnens (Fing- rechnen) . . . . . . . . . . . . . . .
. . . . 7 2. Das Rechnen bei den vom Griechentum unabhangigen
Kulturvolkern. . . . . . . . . . . . . . . 11 3. Das Rechnen bei
den Griechen und Romern 19 4. Oie Abazisten des Fruhmittelalters .
30 5. Das Rechnen bei den Indern . . . . . . . 36 6. Das Rechnen
bei den Arabern . . . . . . . 40 7. Die Auswirkung der
indisch-arabischen Rechenmethoden im Abendland a) das Rechnen mit
ganzen Zahlen. . . . . . . . . 44 . b) das Rechnen mit Bruchen und
die Brfindung der De. zimalbruchrechnung . . . . . 48 8. Die
Ahazisten des Spatmittelalters 51 9. Schluss . . . . . . . . . . .
. . 54 1. DIE ERSTEN ANFANGE DESZAHLENSUND RECHNENS (FINGERRECHNEN)
Die Menschen haben in den altesten Zeiten in rein an schaulicher
Weise mit Hilfe konkreter Gegenstande, z. B. mit Steinchen,
Weizenkornern, Schlangenkopfen, Stabchen, Grashalmen,
Knotenschnnren, Kerbholzern gezahlt und ge rechnet. Ausdrncke wie
das franzosische "calculer" = "mit Steinehen hantieren" und die
Bezeichnung der mexikanischen Tarahumaren fur rechnen=, ab haufen"
oder "zu Haufen ver teilen" weisen noch darauf hin. Das am
weitesten verbreitete und beliebteste Rechenhilfsmittel der
Menschheit waren aber die Finger. Sie wurden benutzt, ahnlich wie
es jetzt noch unsere Kinder in der Schule machen. Belege dafur
lassen sich genug erbringen."
Jesuit engagement with natural philosophy during the late 16th and
early 17th centuries transformed the status of the mathematical
disciplines and propelled members of the Order into key areas of
controversy in relation to Aristotelianism. Through close
investigation of the activities of the Jesuit 'school' of
mathematics founded by Christoph Clavius, The Scientific
Counter-Revolution examines the Jesuit connections to the rise of
experimental natural philosophy and the emergence of the early
scientific societies. Arguing for a re-evaluation of the role of
Jesuits in shaping early modern science, this book traces the
evolution of the Collegio Romano as a hub of knowledge. Starting
with an examination of Clavius's Counter-Reformation agenda for
mathematics, Michael John Gorman traces the development of a
collective Jesuit approach to experimentation and observation under
Christopher Grienberger and analyses the Jesuit role in the Galileo
Affair and the vacuum debate. Ending with a discussion of the
transformation of the Collegio Romano under Athanasius Kircher into
a place of curiosity and wonder and the centre of a global
information gathering network, this book reveals how the
Counter-Reformation goals of the Jesuits contributed to the shaping
of modern experimental science.
Pythagorean thought, from the civilisation of Ancient Greece, is
still prevalent in religion, poetry, philosophy, music,
architecture and the classical sciences today. This fascinating and
insightful collection of essays by experts in their fields explores
the Pythagorean tradition, drawing out connections in form, number
and geometry as well as light, colour, music and poetry. The
contributors include Robert Lawlor, Keith Critchlow, Kathleen
Raine, Anne Macaulay and Arthur Zajonc. Previously published as
Rediscovering Sacred Science.
Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer
Book Archives mit Publikationen, die seit den Anfangen des Verlags
von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv
Quellen fur die historische wie auch die disziplingeschichtliche
Forschung zur Verfugung, die jeweils im historischen Kontext
betrachtet werden mussen. Dieser Titel erschien in der Zeit vor
1945 und wird daher in seiner zeittypischen politisch-ideologischen
Ausrichtung vom Verlag nicht beworben.
An erster Stelle bringt dieses Doppelheft der Mater i a I i e n
einen Abschnitt aus dem in Vorbereitung befindlichen Aufsatze von
A. Ga 11 e uber die geodatischen Arbeiten von Gauss; da es sich um
einen Gegenstand handelt, der uber die Geodasie hinaus greift und
ein selbstandiges Interesse besitzt, schien es angebracht, diesen
Abschnitt .dem eigentlichen Aufsatze vorauszuschicken. Das
Zahlenrechnen greift in die ganze Tatigkeit von Gauss in der reinen
wie in der angewandten Mathematik ein, und darum war dem Verfasser
die gutige Unterstutzung der Mitarbeiter an der Herausgabe der
Werke von Gauss, sowie einiger anderer Gelehrten sehr willkommen;
er mochte auch an dieser Stelle M. Brendel, F. Klein, L. Kruger, A.
Loewy, P. Maennchen, L. Schlesinger, P. Stackel und G. Witt fur
verschiedene Bemerkungen und Hin weise seinen verbindlichen Dank
zum Ausdruck bringen. An zweiter Stelle erscheint der Aufsatz von
P. Stackel uber Gauss als Geometer, der sich den Aufsatzen von Bach
mann uber die zahlentheoretischen und von Schlesinger uber die
funktionentheoretischen Arbeiten von Gauss anreiht. Der Ver fasser
ist F. Klein und L. Schlesinger fur das grosse Interesse
verpflichtet, dass sie seiner Arbeit wahrend ihrer Entstehung ent
gegenbrachten, nicht minder auch fur eine Reihe wertvoller Be
merkungen, die sie wahrend der Korrektur beigesteuert haben. An der
Korrektur beteiligten sich ferner F. Engel, A. Galle und A."
zu der hinterlassenen Abllamllullg VOll Abel, S. 57-81. -- 1 Die
Definition der Ordnung eines algebraischen Ausdrucks, wie sie auf
Seite 67 gegeben ist, ist incorrcct und nach der auf S. 10
angefiihrten zu berichtigen. Die Ordnung eines algebraischen
Ausdrucks ist also nicht gleich der Anzahl der in ihm ausser den
bekannten Groessen auftretenden Wurzelgroessen, sondern vielmehr,
wenn man sich des Symbols V-Wie ublich zur Bezeichnung der
Wurzelgroessen bedient, gleich der groessten von denjenigen Zahlen,
welche angeben, wie viele solcher Wurzelzeichen sich in dem
gegebenen algebraischen Ausdruck uber einander erstrecken. Dabei
wird vorausgesetzt, dass, wenn ein Wurzelzeichen einen Index hat,
welcher eine zusammengesetzte Zahl ist, dasselbe nach der Formel
1Jtn m -V-= VFso weit umgeformt werde, bis siimtliche Wurzelzeiehen
Primzahl exponenten tragen, und dass sich keines dieser
Wurzelzeichen durch Ausfuhrung der durch dasselbe angedeuteten
Operation beseitigen Hisst. Kommen in einem algebraischen Ausdruck
mehrere solcher auf einander oder auf algebrai. che Ausdrucke
niederer Ordnung nicht reducierbarer Wurzelgroessen vor, in denen
jene, die groesste Anzahl der iiber einander sich erstreekenden
'Wurzelzeichen angebenden Zahlen einander gleich sind, so giebt die
Anzahl derselben den Grad des algebraischen Ausdrucks an. - Ist In
die Ordnung des algebraischen Ausdrucks und bezeichnet man die
einzelnen Wurzelgroessen in der Reihenfolge, wie sie numerisch
berechnet werden ter mussen, um den Wert der Wurzelgroesse m
Ordnung zu erhalten, mit ""m-l . . . .
Now available in a one-volume paperback, this book traces the
development of the most important mathematical concepts, giving
special attention to the lives and thoughts of such mathematical
innovators as Pythagoras, Newton, Poincare, and Godel. Beginning
with a Sumerian short story--ultimately linked to modern digital
computers--the author clearly introduces concepts of binary
operations; point-set topology; the nature of post-relativity
geometries; optimization and decision processes; ergodic theorems;
epsilon-delta arithmetization; integral equations; the beautiful
"ideals" of Dedekind and Emmy Noether; and the importance of
"purifying" mathematics. Organizing her material in a conceptual
rather than a chronological manner, she integrates the traditional
with the modern, enlivening her discussions with historical and
biographical detail.
This volume offers a broad, philosophical discussion on mechanical
explanations. Coverage ranges from historical approaches and
general questions to physics and higher-level sciences . The
contributors also consider the topics of complexity, emergence, and
reduction. Mechanistic explanations detail how certain properties
of a whole stem from the causal activities of its parts. This kind
of explanation is in particular employed in explanatory models of
the behavior of complex systems. Often used in biology and
neuroscience, mechanistic explanation models have been often
overlooked in the philosophy of physics. The authors correct this
surprising neglect. They trace these models back to their origins
in physics. The papers present a comprehensive historical,
methodological, and problem-oriented investigation. The
contributors also investigate the conditions for using models of
mechanistic explanations in physics. The last papers make the
bridge from physics to economics, the theory of complex systems and
computer science . This book will appeal to graduate students and
researchers with an interest in the philosophy of science,
scientific explanation, complex systems, models of explanation in
physics higher level sciences, and causal mechanisms in science.
During the period of national isolation, a mathematical tradition
called wasan flourished in Japan. Though virtually unknown to
Europeans before the Meiji Restoration, its practitioners, the
wasanka, produced some results comparable to (and sometimes in
advance of) those of mathematicians of the European Enlightment.
This volume, a companion to Unger's earlier translation of
solutions by Aida Yasuaki (1747-1817), focuses on problems that
Aida most likely used as a teacher. Unger explains the reasons for
believing this, and sheds further light on the intellectual milieu
in which Aida worked by discussing other books by Aida, including
one in which he describes Dutch techniques of navigation.
Alain Badiou has claimed that Quentin Meillassoux's book After
Finitude (Bloomsbury, 2008) "opened up a new path in the history of
philosophy." And so, whether you agree or disagree with the
speculative realism movement, it has to be addressed. Lacanian
Realism does just that. This book reconstructs Lacanian dogma from
the ground up: first, by unearthing a new reading of the Lacanian
category of the real; second, by demonstrating the political and
cultural ingenuity of Lacan's concept of the real, and by
positioning this against the more reductive analyses of the concept
by Slavoj Zizek, Alain Badiou, Saul Newman, Todd May, Joan Copjec,
Jacques Ranciere, and others, and; third, by arguing that the
subject exists intimately within the real. Lacanian Realism is an
imaginative and timely exploration of the relationship between
Lacanian psychoanalysis and contemporary continental philosophy.
The first critical work to attempt the mammoth undertaking of
reading Badiou's Being and Event as part of a sequence has often
surprising, occasionally controversial results. Looking back on its
publication Badiou declared: "I had inscribed my name in the
history of philosophy". Later he was brave enough to admit that
this inscription needed correction. The central elements of
Badiou's philosophy only make sense when Being and Event is read
through the corrective prism of its sequel, Logics of Worlds,
published nearly twenty years later. At the same time as presenting
the only complete overview of Badiou's philosophical project, this
book is also the first to draw out the central component of
Badiou's ontology: indifference. Concentrating on its use across
the core elements Being and Event-the void, the multiple, the set
and the event-Watkin demonstrates that no account of Badiou's
ontology is complete unless it accepts that Badiou's philosophy is
primarily a presentation of indifferent being. Badiou and
Indifferent Being provides a detailed and lively section by section
reading of Badiou's foundational work. It is a seminal source text
for all Badiou readers.
In recent years there have been a number of books-both anthologies
and monographs-that have focused on the Liar Paradox and, more
generally, on the semantic paradoxes, either offering proposed
treatments to those paradoxes or critically evaluating ones that
occupy logical space. At the same time, there are a number of
people who do great work in philosophy, who have various semantic,
logical, metaphysical and/or epistemological commitments that
suggest that they should say something about the Liar Paradox, yet
who have said very little, if anything, about that paradox or about
the extant projects involving it. The purpose of this volume is to
afford those philosophers the opportunity to address what might be
described as reflections on the Liar.
The term "fuzzy logic" (FL), as it is understood in this book,
stands for all aspects of representing and manipulating knowledge
based on the rejection of the most fundamental principle of
classical logic: the principle of bivalence. According to this
principle, each declarative sentence is required to be either true
or false. In fuzzy logic, these classical truth values are not
abandoned. However, additional, intermediary truth values between
true and false are allowed, which are interpreted as degrees of
truth. This opens a new way of thinking-thinking in terms of
degrees rather than absolutes. For example, it led to the
definition of a new category of sets, referred to as fuzzy sets, in
which membership is a matter of degree. The book examines the
genesis and development of fuzzy logic. It surveys the prehistory
of fuzzy logic and inspects circumstances that eventually lead to
the emergence of fuzzy logic. The book explores in detail the
development of propositional, predicate, and other calculi that
admit degrees of truth, which are known as fuzzy logic in the
narrow sense. Fuzzy logic in the broad sense, whose primary aim is
to utilize degrees of truth for emulating common-sense human
reasoning in natural language, is scrutinized as well. The book
also examines principles for developing mathematics based on fuzzy
logic and provides overviews of areas in which this has been done
most effectively. It also presents a detailed survey of established
and prospective applications of fuzzy logic in various areas of
human affairs, and provides an assessment of the significance of
fuzzy logic as a new paradigm.
We live an information-soaked existence - information pours into
our lives through television, radio, books, and of course, the
Internet. Some say we suffer from 'infoglut'. But what is
information? The concept of 'information' is a profound one, rooted
in mathematics, central to whole branches of science, yet with
implications on every aspect of our everyday lives: DNA provides
the information to create us; we learn through the information fed
to us; we relate to each other through information transfer -
gossip, lectures, reading. Information is not only a mathematically
powerful concept, but its critical role in society raises wider
ethical issues: who owns information? Who controls its
dissemination? Who has access to information? Luciano Floridi, a
philosopher of information, cuts across many subjects, from a brief
look at the mathematical roots of information - its definition and
measurement in 'bits'- to its role in genetics (we are
information), and its social meaning and value. He ends by
considering the ethics of information, including issues of
ownership, privacy, and accessibility; copyright and open source.
For those unfamiliar with its precise meaning and wide
applicability as a philosophical concept, 'information' may seem a
bland or mundane topic. Those who have studied some science or
philosophy or sociology will already be aware of its centrality and
richness. But for all readers, whether from the humanities or
sciences, Floridi gives a fascinating and inspirational
introduction to this most fundamental of ideas. ABOUT THE SERIES:
The Very Short Introductions series from Oxford University Press
contains hundreds of titles in almost every subject area. These
pocket-sized books are the perfect way to get ahead in a new
subject quickly. Our expert authors combine facts, analysis,
perspective, new ideas, and enthusiasm to make interesting and
challenging topics highly readable.
Generality is a key value in scientific discourses and practices.
Throughout history, it has received a variety of meanings and of
uses. This collection of original essays aims to inquire into this
diversity. Through case studies taken from the history of
mathematics, physics and the life sciences, the book provides
evidence of different ways of understanding the general in various
contexts. It aims at showing how collectives have valued generality
and how they have worked with specific types of "general" entities,
procedures, and arguments. The books connects history and
philosophy of mathematics and the sciences at the intersection of
two of the most fruitful contemporary lines of research: historical
epistemology, in which values (e.g. "objectivity", "accuracy") are
studied from a historical viewpoint; and the philosophy of
scientific practice, in which conceptual developments are seen as
embedded in networks of social, instrumental, and textual
practices. Each chapter provides a self-contained case-study, with
a clear exposition of the scientific content at stake. The
collection covers a wide range of scientific domains - with an
emphasis on mathematics - and historical periods. It thus allows a
comparative perspective which suggests a non-linear pattern for a
history of generality. The introductory chapter spells out the key
issues and points to the connections between the chapters.
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