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Das Kalenderblatt 110105

04/01/2011 - 12:02 von WM | Report spam
Krieg der Frösche und der Màuse (33)

Meanwhile, under the pressure of well-founded criticism exerted upon
old formalism, Hilbert founded the New Formalist School, which
postulated existence and exactness independent of language not for
proper mathematics but for meta-mathematics, which is the scientific
consideration of the symbols occurring in perfected mathematical
language, and of the rules of manipulation of these symbols. On this
basis new formalism, in contrast to old formalism, in confesso made
primordial practical use of the intuition of natural numbers and of
complete induction. It is true that only for a small part of
mathematics (much smaller than in pre-intuitionism) was autonomy
postulated in this way. New formalism was not deterred from its
procedure by the objection that between the perfection of mathematical
language and the perfection of mathematics itself no clear connection
could be seen.

So the situation left by formalism and pre-intuitionism can be
summarised as follows: for the elementary theory of natural numbers,
the principle of complete induction and more or less considerable
parts of arithmetic and of algebra, exact existence, absolute
reliability and non-contradictority were universally acknowledged,
independently of language and without proof. As for the continuum, the
question of its languageless existence was neglected, its
establishment as a set of real numbers with positive measure was
attempted by logical means and no proof of its non-contradictory
existence appeared. For the whole of mathematics the four principles
of classical logic were accepted as means of deducing exact truths.

In this situation intuitionism intervened with two acts, of which the
first seems to lead to destructive and sterilising consequences, but
then the second yields ample possibilities for new developments.

[L.E.J. Brouwer: "Lectures on Intuitionism - Historical Introduction
and Fundamental Notions" (1951), Cambridge University Press (1981)]

Gruß, WM
 

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#1 Carsten Schultz
04/01/2011 - 12:28 | Warnen spam
Am 04.01.11 12:02, schrieb WM:

Krieg der Frösche und der Màuse (33)

Meanwhile, under the pressure of well-founded criticism exerted upon
old formalism, Hilbert founded the New Formalist School, which
postulated existence and exactness independent of language not for
proper mathematics but for meta-mathematics, which is the scientific
consideration of the symbols occurring in perfected mathematical
language, and of the rules of manipulation of these symbols. On this
basis new formalism, in contrast to old formalism, in confesso made
primordial practical use of the intuition of natural numbers and of
complete induction. It is true that only for a small part of
mathematics (much smaller than in pre-intuitionism) was autonomy
postulated in this way. New formalism was not deterred from its
procedure by the objection that between the perfection of mathematical
language and the perfection of mathematics itself no clear connection
could be seen.

So the situation left by formalism and pre-intuitionism can be
summarised as follows: for the elementary theory of natural numbers,
the principle of complete induction and more or less considerable
parts of arithmetic and of algebra, exact existence, absolute
reliability and non-contradictority were universally acknowledged,
independently of language and without proof. As for the continuum, the
question of its languageless existence was neglected, its
establishment as a set of real numbers with positive measure was
attempted by logical means and no proof of its non-contradictory
existence appeared. For the whole of mathematics the four principles
of classical logic were accepted as means of deducing exact truths.

In this situation intuitionism intervened with two acts, of which the
first seems to lead to destructive and sterilising consequences, but
then the second yields ample possibilities for new developments.

[L.E.J. Brouwer: "Lectures on Intuitionism - Historical Introduction
and Fundamental Notions" (1951), Cambridge University Press (1981)]



Tu uns doch den Gefallen und tippe auch noch die drei entsprechenden
Absàtze seiner Berliner Gastvorlesung (1927) ab, die sind auf deutsch
und sprachlich interessanter:

Brouwer, Luitzen Egbertus Jan
Intuitionismus. Hrsg., eingel. und komment. von Dirk van Dalen.
Mannheim: B.I.-Wissenschaftsverlag. 161 S. (1992). ISBN 3-411-15371-7


Carsten Schultz (2:38, 33:47)
http://carsten.codimi.de/
PGP/GPG key on the pgp.net key servers,
fingerprint on my home page.

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