Das Kalenderblatt 091202

01/12/2009 - 13:01 von WM | Report spam
First, there is a major misunderstanding from which many
misinterpretations follow. {{Das passiert des öftere auch
andernorts.}} We have the impression that Gold didn't really get the
main thesis of our book: It is the human embodied mind that brings
mathematics into being. {{Und ohne jenen ist diese nicht! Insbesondere
gibt es also keine Absonderlichkeiten wie unkennbare Zahlen sondern
allenfalls den absonderlichen Gedanken, dass solche Absonderlichkeiten
zur Mathematik gehören könnten.}} This is precisely what the subtitle
of the book explicitly indicates. "Embodiment" thus, with its strong
biological and cognitive constraints, is a fundamental theoretical
component that gives shape and continuity to the entire book. It is
under this view that the cognitive mechanisms we describe, among which
conceptual metaphor is one of the most important ones, make sense at
all: Conceptual metaphor has explanatory power precisely because they
are empirically observable embodied mechanisms that satisfy strong
biological and cognitive constraints. The term "embodiment" and its
related concepts appear all over the book, to the point that both
summarizing theoretical chapters at the end are entitled "The Theory
of Embodied Mathematics" (Chapter 15) and "The Philosophy of Embodied
Mathematics" (Chapter 16). In Gold's review, however, the term
embodiment is not even mentioned once!

This is not just picky terminology. Understanding the role of
embodiment and its biological and cognitive constraints {{and the
physical constraints, one might add}} is extremely important to get an
idea of what our book is about. So when Gold refers to conceptual
metaphors as being "essentially isomorphisms" she presents an
inaccurate picture, which leads her to miss some fundamental points of
the book, as well as the continuity and overall structure of it. [...]
It is simply a confusion between disciplines to refer to conceptual
metaphors as "isomorphisms." Our book is within the discipline of
cognitive science and its subject matter is the cognitive science of
mathematical ideas. To refer to conceptual metaphors as isomorphisms
is to assume that the book is within the discipline of mathematics,
which it is not. Our book is an attempt to give an account of
mathematical ideas and inferences in terms of biologically and
cognitively plausible mechanisms of the human mind, such as conceptual
metaphors.

For example, the conceptual metaphor Arithmetic As Object Collection
to which Gold refers is not a mere descriptive isomorphism. It is the
embodied cognitive mechanism that gives an account of why the
empirically observed expressions that exist in human (even technical)
communication such as "three is bigger than two" or "four is smaller
than eight" have the precise meaning they have, despite the fact that
numbers in themselves don't have size. We believe that, because Gold
misses the deep role of embodiment in our theoretical account
throughout the book {{sie ist leider nicht die einzige.}}

We would also like to clarify a couple of technical details regarding
how conceptual metaphors work. Gold says, correctly, that the Basic
Metaphor of Infinity (BMI) is the most important metaphor in the book,
and she points out (also correctly) that the BMI is not an isomorphism
(that's right!). Gold accurately observes that the Basic Metaphor of
Infinity characterizes "something genuinely new" (i.e., an end to an
unending process: actual infinity). Unfortunately, she seems not to
understand what conceptual metaphors are and how they differ in kind
from disembodied mathematical isomorphisms (which are literal, not
metaphorical). As a result, she mistakenly claims that the BMI
introduces an "ambiguity of how to go from the intermediate states to
the final state", leaving "a gap that needs more explanation." It is
incorrectly taking conceptual metaphors to be mathematical
isomorphisms that generate that gap. Conceptual metaphors, being human
cognitive mechanisms have many properties not captured by
isomorphisms. As we say it explicitly in pages 45 and 46, "conceptual
metaphors do not just map preexisting elements of the source domain
onto preexisting elements of the target domain. They can also
introduce new elements into the target domain" (italics in the book).
These elements are not inherent to the target domain. [...]

The moral here is this: It is totally consistent with what we know
about human cognitive mechanisms that actual infinity could be a
metaphorical idea. Via a specific conceptual metaphor (the BMI), an
unending iterative process that goes on and on can be conceptualized
as a process with an actual end and an actual final resultant state.
{{Ja, leider ist es völlig consistent mit dem, was wir über die
Menschen wissen.}}

Reply to Bonnie Gold's review of "Where Mathematics Comes From: How
the Embodied Mind Brings Mathematics into Being" by George Lakoff and
Rafael E. Núñez (2001).
http://www.maa.org/reviews/wheremath_reply.html

Gruß, WM
 

Lesen sie die antworten

#1 Ralf Bader
02/12/2009 - 01:15 | Warnen spam
WM wrote:

First, there is a major misunderstanding from which many
misinterpretations follow. {{Das passiert des öftere auch
andernorts.}} We have the impression that Gold didn't really get the
main thesis of our book: It is the human embodied mind that brings
mathematics into being. {{Und ohne jenen ist diese nicht! Insbesondere
gibt es also keine Absonderlichkeiten wie unkennbare Zahlen sondern
allenfalls den absonderlichen Gedanken, dass solche Absonderlichkeiten
zur Mathematik gehören könnten.}}


[...]
This is not just picky terminology. Understanding the role of
embodiment and its biological and cognitive constraints {{and the
physical constraints, one might add}} is extremely important to get an
idea of what our book is about.


[...]
We believe that, because Gold
misses the deep role of embodiment in our theoretical account
throughout the book {{sie ist leider nicht die einzige.}}

We would also like to clarify a couple of technical details regarding
how conceptual metaphors work. Gold says, correctly, that the Basic
Metaphor of Infinity (BMI) is the most important metaphor in the book,
and she points out (also correctly) that the BMI is not an isomorphism
(that's right!). Gold accurately observes that the Basic Metaphor of
Infinity characterizes "something genuinely new" (i.e., an end to an
unending process: actual infinity).


[...]
The moral here is this: It is totally consistent with what we know
about human cognitive mechanisms that actual infinity could be a
metaphorical idea. Via a specific conceptual metaphor (the BMI), an
unending iterative process that goes on and on can be conceptualized
as a process with an actual end and an actual final resultant state.
{{Ja, leider ist es völlig consistent mit dem, was wir über die
Menschen wissen.}}


[...]
Reply to Bonnie Gold's review of "Where Mathematics Comes From: How
the Embodied Mind Brings Mathematics into Being" by George Lakoff and
Rafael E. Núñez (2001).
http://www.maa.org/reviews/wheremath_reply.html

Gruß, WM



Die "cognitive science of mathematics" müßte natürlich auch etwas dazu sagen
können, weshalb manche Menschen offenbar nicht fàhig sind, "the most
important metaphor in the book" nachzuvollziehen. Und die
allgemeine "cognitive science" etwas zu dem offenbar möglichen Phànomen,
daß jemand alles, was ihm begegnet, als Bestàtigung einer
selbstausgebrüteten abstrusen Pseudotheorie fehlinterpretiert, auch wenn es
dieser offenbar diametral entgegensteht. Wie es also geschehen kann, daß
die menschlichen kognitiven Fàhigkeiten in Wahn abgleiten. Insofern ist
auch mathematischer Crackpotismus ein interessantes Studienobjekt.

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