Das Kalenderblatt 091201

30/11/2009 - 19:13 von WM | Report spam
Lakoff and Núñez's avowed purpose is to begin laying the
foundations for a truly scientific understanding of mathematics, one
grounded in processes common to all human cognition. They find that
four distinct but related processes metaphorically structure basic
arithmetic: object collection, object construction, using a measuring
stick, and moving along a path.
[...] Lakoff and Núñez hold that mathematics results from the human
cognitive apparatus and must therefore be understood in cognitive
terms. WMCF {{Where Mathematics Comes From}} advocates (and includes
some examples of) a cognitive idea analysis of mathematics which
analyzes mathematical ideas in terms of the human experiences,
metaphors, generalizations, and other cognitive mechanisms giving rise
to them. Idea analysis is distinct from mathematics and cannot be
performed by mathematicians unless they are trained in cognitive
Lakoff and Núñez start by reviewing the psychological literature,
concluding that humans appear to have an innate ability, called
subitizing, to count, add, and subtract up to about 4 or 5. They
document this conclusion by reviewing the literature, published in
recent decades, describing experiments with infant subjects. For
example, infants quickly become excited or curious when presented with
"impossible" situations, such as having three toys appear when only
two were initially present.
The authors argue that mathematics goes far beyond this very
elementary level thanks to a large number of metaphorical
constructions. For example, they argue that the Pythagorean position
that all is number, and the associated crisis of confidence that came
about with the discovery of the irrationality of the square root of
two, arises solely from a metaphorical relation between the length of
the diagonal of a square, and the possible numbers of objects.
Much of WMCF deals with the important concepts of infinity and of
limit processes, seeking to explain how finite humans living in a
finite world could eventually conceive of the actual infinite. Thus
much of WMCF is, in effect, a study of the epistemological foundations
of the calculus. Lakoff and Núñez conclude that while the potential
infinite is not metaphorical, the actual infinite is. Moreover, they
deem all manifestations of actual infinity to be instances of what
they call the "Basic Metaphor of Infinity."
WMCF emphatically rejects the Platonistic philosophy of
mathematics. They emphasize that all we know and can ever know is
human mathematics, the mathematics arising from the human intellect.
Whether a transcendent mathematics, independent of human thought, can
be said to exist is probably an unanswerable question, and perhaps
even a meaningless one.

George Lakoff, Rafael E. Núñez: "Where Mathematics Comes From: How the
Embodied Mind Brings Mathematics into Being", Basic Books (2000)

Gruß, WM

Lesen sie die antworten

#1 Peter
30/11/2009 - 20:55 | Warnen spam
On 30 Nov., 19:13, WM wrote:
   Lakoff and Núñez's avowed purpose is to begin laying the

Sie könnten durchaus etwas klarer zum Ausdruck bringen, woher dieser
Text stammt, insbesondere wenn zwei andere Referenzen dem
hinweisenden Link vorangehen.

Besser als die zitierte schülerhaft-schwafelige Besprechung von
Wikipedia finde ich Zusammenfassung von Rafael Núñez in seinem
Aufsatz "Do Real Numbers Really Move?" [1].

| From the perspective of the cognitive science of mathematics
| a different view emerges: Mathematics doesn't exist outside of
| human cognition.
| Formal definitions and axioms in mathematics are themselves created
| by human ideas (although they constitute a very small and specific
| fraction of human cognition), and they only capture very limited
| aspects of the richness of mathematical ideas.

Sofern man nicht durch idealistische deutsche Gymnasiallehrer
philosophisch missgebildet ist, erscheint einem eine solche Sicht
als naheliegend, wenn nicht gar als selbstverstàndlich.

Das eigentliche Interessante ist, /wie/ hier Einsichten in die
Gedankenwelt der Mathematik gesucht wird: auf empirischen Weg
mit "Gesture Studies", einer ganz jungen Disziplin [2][3].

Núñez filmt unter anderem Mathematiker bei ihren Vorlesungen
und analysiert dann Körperbewegungen, Gesichtsausdruck und
Gesten und korreliert dies mit selbst so abstrakten Dingen wie
dem Existenz- oder Allquantor.

Es ist sicher noch viel zu früh zu beurteilen, wie weit das alles
tràgt, aber ein faszinierender Zugang ist es allemal.

P. S. Einige der Filme, die Núñez in seinen Studien ausgewertet
hat, sind folgende Vorlesungen von Richard Feynman[4], auf die
ich kürzlich schon auf dsp hingewiesen habe.

[1] Reuben Hersh (ed.), Unconventional essays on the
nature of mathematics. Springer 2006.
[2] Man google z. B. nach
"Gesture: Evolution, brain and linguistic structures"
[3] http://www.mpi.nl/
[4] http://tiny.cc/PqK1o
The Messenger Lectures include seven videos of
Dr. Richard Feynman speaking on physiscs

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