Встреча

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	К	Пуденко Сергей
	Дата	31.05.2008 08:56:26
	Рубрики	Прочее; Наука & природа;

Базовая метафора Бесконечности

Книга,конечно опять непереведенная, и практически в ибане не известная(в Ибанске вчоным такое не трэба)
Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into
Being. New York: Basic Books, 2000.

Кто тут интересовался Матураной-Варелой,кажется А.Михайлов?

Вышла циклопедия, на натахаусе. Полный текст с картинками ссписал у Косиловой
Величковский Б. - Когнитивная наука. Основы психологии познания. В 2-х томах.2006 Том II.doc

про Лакоффа там кратко

для освежения памяти кратко пробегом
http://www.philology.ru/linguistics1/drulak-06.htm
образная рациональность метафор представляет идеальную общую основу для сведения воедино рационального познания и художественного переживания. В настоящее время метафоры обычно рассматриваются либо применительно к науке [Lakoff, Nunez 2000], либо применительно к искусству [Lakoff, Turner 1989]. Тем не менее, редко рассматриваются метафоры из научных текстов и художественных работ с целью изучения того, как эти два способа нашего познания мира могут быть соединены в рамках одного метода.

Проблемы сязаны с преподавание математики в школах.
в Венесуэле и той плотно занимаются. Спецномер
http://psb.sbras.ru/EMIS/journals/BAMV/vol10.html#numero2
Todo/All Volumen 10, num. 1 (pdf 2.2 MB)
Nu'mero 2 - Number 2
Edicio'n Especial

есть пдфы на испанском
Educacio'n Matema'tica
Matema'ticas y Cosas. Una Mirada Desde la Educacio'n Matema'tica
Vicenc Font, p. 249 (pdf)

Where Mathematics Comes From: ответ Лакоффа на критику его концепции Пинкером(ортодоксальным хомскианцем)

Новое представление заключается в том, что разум имеет материаль-
ное воплощение. Мозг вызывает мысли в виде концептуальных рамок,
образов-схем, прототипов, концептуальных метафор и концептуаль-
ных смесей. Процесс мышления — это не алгоритмическая манипуля-
ция символами, а скорее нейрональное вычисление с использованием
механизмов мозга. Эти механизмы рассматриваются в недавней книге
Джерома Фельдмана «От молекул к метафорам».
Вопреки Декарту мышление использует именно такие механизмы,
а не формальную логику. Мышление преимущественно бессознательно,
и, как писал Антонио Дамасио в «Ошибке Декарта», рациональность
требует эмоций.

Базовая метафора Бесконечности
Part III
THE EMBODIMENT OF INFINITY
8 The Basic Metaphor of Infinity 155
9 Real Numbers and Limits 181
vii
0465037704fm.qxd 8/23/00 9:49 AM Page vii
10 Transfinite Numbers 208
11 Infinitesimals

Сайт Нуньеса с представлением книги
http://perso.unifr.ch/rafael.nunez/booktalks.html

http://www.mprof.ru/eng/index.php?lp=ru_en&trurl=http%3a%2f%2fen.wikipedia.org%2fwiki%2fGeorge_Lakoff

Ответ на рецензию в амер.мат обществе
http://www.maa.org/reviews/wheremath_reply.html
http://www.maa.org/reviews/images/wheremath.jpg

According to Lakoff and Nґu˜nez [1], one of the most important and the most impressive
metaphors in mathematics is the BMI, or the Basic Metaphor of Infinity.
BMI is the metaphor which changes potential infinity into actual infinity. Given a
situation where some operation continues endlessly, BMI will form the conceptual situation
where the operation ‘has been repeated an infinite number of times’.
is not a book of mathematics. It is a book about mathematics. It is a
book on mathematical idea analysis. Inside mathematics, there is a rule that something
proved from axioms using logical manipulations is called a theorem. Mathematical idea
analysis tries to explain why the theorem is true, not by the proof, but by the ‘meaning’
of that theorem. The reason that a theorem is true is not because that theorem can be
proven based on the ZFC axioms (that is, there exists a proof), but because it represents
a content meaningful for human beings.
Mathematics is a creation of the human brain, and mathematical idea analysis can
explain why some facts had to be treated as a theorem by human mathematics. In
mathematics, if there is a theorem hard to prove, mathematicians change the axioms or
change the definitions to somehow prove it. By doing so, mathematicians have extended
the world of mathematics. Then, what is the mathematical world that they want to
extend, paying that much effort? The answer lies not outside the fact that human beings
live with human brains.

The most interesting and fully developed example of a conceptual metaphor is the 'Basic Metaphor of Infinity'.

Lakoff and Nъсez have introduced the so-called Basic Metaphor of Infinity (BMI),
which arises when one conceptualises actual infinity as the result of an iterative
process (Lakoff & Nщсez, 2000, p. 159). The two domains (source and target) of the
metaphor are characterised by an ordinary iterative process with an indefinite number
of iterations, each of which has an initial state and a resultant state. The crucial effect
of the metaphor is to add to the target domain the completion of the process and its
resultant state as a unique final state. This metaphor allows to conceptualise infinity
in terms of the unique and final result of a process (Lakoff and Nщсez, 2000, p.160):
Via the BMI, infinity is converted from an open-ended process to a specific, unique entity.
Lakoff and Nщсez point out some important general features in the conceptualisation
of the infinity.

http://books.google.ru/books?id=AQZ5jmmpaDAC&pg=PA113&lpg=PA113&dq=lakoff+%22The+Basic+Metaphor+of+Infinity%22&source=web&ots=sMlw7q6ZJl&sig=HcKxIDexYFqck6V4KImNJXhoRwg&hl=ru
Handbook of Mathematical Cognition
New York : Psychology Press
J. Campbell (Ed.).
- 9 -
At least since the time of Aristotle, there have been two concepts of infinity, potential
infinity and actual infinity. Suppose you start to count: 1, 2, 3, … and you imagine you
go on indefinitely without stopping. That is an instance of potential infinity, infinity
without an end. On the other hand consider the set of all natural numbers. No one could
ever enumerate all of them; the enumeration would go on without end. Yet we
conceptualize a set containing all of them, even though the enumeration has never and
could never produce them all. That is an instance of actual infinity — an infinite
completed thing!
In Where Mathematics Comes From we hypothesize that the idea of “actual
infinity” in mathematics is metaphorical, that all the diverse ideas using actual infinity
make use of the ultimate metaphorical result of a process without end. Literally, there is
no such thing; if the process does not end, there can be no such “ultimate result.” But the
very human mechanism of metaphor allows us to conceptualize and construct the “result”
of an infinite process — in terms of the only way we have for conceptualizing the result
of a process—in terms of a process that does have an end.
We hypothesize that all cases of actual infinity —from infinite sets to points at
infinity to limits of infinite series to infinite intersections to least upper bounds—are
special cases of a single general conceptual metaphor in which processes that go on
indefinitely (that is, without end) are conceptualized as having an end and an ultimate
result. We call this metaphor the BASIC METAPHOR OF INFINITY — or the BMI for short
(Lakoff & Nъсez, 2000. For details regarding how the BMI applies to Georg Cantor’s
transfinite cardinal numbers, see Nъсez, in press).
Nъсez, R.(in press). Creating Mathematical Infinities: The Beauty of Transfinite
Cardinals. Journal of Pragmatics.
Nъсez, R. (in preparation). Inferential Statistics in the Context of Empirical Cognitiv
Lakoff
http://www.cogsci.ucsd.edu/~nunez/COGS260/Nunez_Lakoff_Hdbk.pdf

недавнее -попытка стыковать БМИ с теорией Кантора
Creating mathematical infinities: Metaphor, blending, and the beauty of transfinite cardinals R.E. Nunez pp 1717-1741

Creating mathematical infinities: Metaphor, blending, and the beauty of transfinite cardinals

Rafael E. Nu'n~ezE-mail The Corresponding Author

Department of Cognitive Science, University of California, San Diego, La Jolla, CA 92093-0515, USA

Received 23 February 2003;
revised 19 September 2003;
accepted 21 September 2004.
Available online 23 May 2005.

Purchase the full-text article
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6VCW-4G7JXR5-1&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=6d8ac65fb62947174d5ea9c1e7da92ce

Creating mathematical infinities: Metaphor, blending, and the beauty of transfinite cardinals

Abstract

The infinite is one of the most intriguing, controversial, and elusive ideas in which the human mind has ever engaged. In mathematics, a particularly interesting form of infinity—actual infinity—has gained, over centuries, an extremely precise and rich meaning, to the point that it now lies at the very core of many fundamental fields such as calculus, fractal geometry, and set theory. In this article I focus on a specific case of actual infinity, namely, transfinite cardinals, as conceived by one of the most imaginative and controversial characters in the history of mathematics, the 19th century mathematician Georg Cantor (1845–1918). The analysis is based on the Basic Metaphor of Infinity (BMI). The BMI is a human everyday conceptual mechanism, originally outside of mathematics, hypothesized to be responsible for the creation of all kinds of mathematical actual infinities, from points at infinity in projective geometry to infinite sets, to infinitesimal numbers, to least upper bounds [Lakoff, George, Nu'n~ez, Rafael, 2000. Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. Basic Books, New York]. In this article I analyze the BMI in terms of a non-unidirectional mapping: a double-scope conceptual blend. Under this view “BMI” becomes the Basic Mapping of Infinity.

Keywords: Metaphor; Mathematics; Blends; Infinity; Transfinite cardinals
Journal of Pragmatics
Volume 37, Issue 10, October 2005, Pages 1717-1741
Conceptual Blending Theory

еще ссылки
http://www.cogsci.ucsd.edu/~nunez/COGS260/Nunez_Lakoff_Hdbk.pdf

http://www.t-kougei.ac.jp/research/pdf/vol1-28-13.pdf
http://www.t-kougei.ac.jp/research/pdf/vol2-27-06.pdf

http://www.ams.org/notices/200110/rev-madden.pdf
http://www.cogsci.ucsd.edu/~nunez/web/CogSci04_paper.pdf
http://www.emis.de/proceedings/PME28/RR/RR115_Robutti.pdf
http://cerme4.crm.es/Papers%20definitius/1/acevedo.pdf