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Matheology § 209
Posted:
Feb 4, 2013 4:23 AM


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 too.}}
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 (2009)] http://plato.stanford.edu/entries/weyl/index.html
Regards, WM



