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Topic: Matheology § 209
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Posts: 18,076
Registered: 1/29/05
Matheology § 209
Posted: Feb 4, 2013 4:23 AM
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Matheology § 209

In Das Kontinuum Weyl says:

The states of affairs with which mathematics deals are, apart from the
very simplest ones, so complicated that it is practically impossible
to bring them into full givenness in consciousness and in this way to
grasp them completely.

Nevertheless, Weyl felt that this fact, inescapable as it might be,
could not justify extending the bounds of mathematics to embrace
notions, such as the actual infinite, which cannot be given fully in
intuition even in principle. He held, rather, that such extensions of
mathematics into the transcendent are warranted only by the fact that
mathematics plays an indispensable role in the physical sciences, in
which intuitive evidence is necessarily transcended. As he says in The
Open World:

? if mathematics is taken by itself, one should restrict oneself with
Brouwer to the intuitively cognizable truths ? nothing compels us to
go farther. But in the natural sciences we are in contact with a
sphere which is impervious to intuitive evidence; here cognition
necessarily becomes symbolical construction. Hence we need no longer
demand that when mathematics is taken into the process of theoretical
construction in physics it should be possible to set apart the
mathematical element as a special domain in which all judgments are
intuitively certain; from this higher standpoint which makes the whole
of science appear as one unit, I consider Hilbert to be right {{me

Weyl soon grasped the significance of Hilbert's program, and came to
acknowledge its ?immense significance and scope?. Whether that program
could be successfully carried out was, of course, still an open
question. But independently of this issue Weyl was concerned about
what he saw as the loss of content resulting from Hilbert's
thoroughgoing formalization of mathematics. ?Without doubt,? Weyl
warns, ?if mathematics is to remain a serious cultural concern {{a
mathematician should never give up this premise}}, then some sense
must be attached to Hilbert's game of formulae.?

[John L. Bell: "Hermann Weyl", Stanford Encyclopedia of Philosophy

Regards, WM

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