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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