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)]

http://www.marxists.org/reference/s...rouwer.htm
Gruß, WM

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