Das Kalenderblatt 110112

11/01/2011 - 07:52 von WM | Report spam
Krieg der Frösche und der Màuse (40)

As long as mathematics was considered as the science of space and
time, it was a beloved field of activity of this classical logic, not
only in the days when space and time were believed to exist
independently of human experience, but still after they had been taken
for innate forms of conscious exterior human experience. There
continued to reign some conviction that a mathematical assertion is
either false or true, whether we know it or not, and that after the
extinction of humanity mathematical truths, just as laws of nature,
will survive. About half a century ago this was expressed by the great
French mathematician Charles Hermite in the following words: 'Il
existe, si je ne me trompe, tout un monde qui est l'ensemble des
vérités mathématiques, dans lequel nous n'avons d'accés que par
l'intelligence, comme existe le monde des réalités physiques [...]"
Only after mathematics had been recognized as an autonomous
interior constructional activity which, although it can be applied to
an exterior world, neither in its origin nor in its methods depends on
an exterior world, firstly all axioms became illusory, and secondly
the criterion of truth or falsehood of a mathematical assertion was
confined to mathematical activity itself, without appeal to logic or
to hypothetical omniscient beings. An immediate consequence was that
for a mathematical assertion a the two cases of truth and falsehood,
formerly exclusively admitted, were replaced by the following three:
(1) a has been proved to be true;
(2) a has been proved to be absurd;
(3) a has neither been proved to be true nor to be absurd, nor do
we know a finite algorithm leading to the statement either that a is
true or that a is absurd. [The case that a has neither been proved to
be true nor to be absurd, but that we know a finite algorithm leading
to the statement either that a is true, or that a is absurd, obviously
is reducible to the first and second cases. This applies in particular
to assertions of possibility of a construction of bounded finite
character in a finite mathematical system, because such a construction
can be attempted only in a finite number of particular ways, and each
attempt proves successful or abortive in a finite number of steps.]
In contrast to the perpetual character of cases (1) and (2), an
assertion of type (3) may at some time pass into another case, not
only because further thinking may generate an algorithm accomplishing
this passage, but also because in modern or intuitionistic
mathematics, as we shall see presently, a mathematical entity is not
necessarily predeterminate, and may, in its state of free growth, at
some time acquire a property which it did not possess before.

[L.E.J. Brouwer: "Changes in the relation between classical logic and
mathematics" (1951), Cambridge University Press (1981)]

Gruß, WM

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#1 Vogel
12/01/2011 - 04:26 | Warnen spam
WM wrote in news:53d1ec19-9b24-436c-9d84-

ist dir schon aufgefallen dass dies hier eine deutschsprachige
Diskussionsgruppe ist?

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