[FOM] What would Weyl do?

[Stephen Pollard]

Here are some observations and speculations about how Hermann Weyl might
have reacted to Harvey Friedman's results.

In THE CONTINUUM, Weyl declares it "entirely certain" that impredicative
set theories (including Zermelo set theory) are "permeated by the poison of
contradiction." I take it that this is a colorful way of saying that they
are inconsistent. As long as he held this extreme view, Weyl would have
insisted that the consequences of classical set theories are irrelevant to
a rational assessment of them. After all, Weyl already "knew" that these
theories imply everything. (If, as he sometimes suggested, these theories
are "clearly meaningless," then their consequences are irrelevant because
they have none. Instead of implying everything, they imply nothing.)

Weyl's mature position was that impredicative set theories may be formally
consistent, but "go beyond such statements as can claim real meaning and
truth founded on evidence." ("Evidence" here is Evidenz or "evidentness":
the experience that a proposition's truth is evident). Weyl himself
observed that it is not always evident what is evident. In borderline cases
where a theory's epistemic status is unclear, he might have accepted
utility or fruitfulness as a reason for employing the theory.

In THE CONTINUUM, Weyl employs utility as a criterion for choosing between
predicative theories. In his search for a reliable foundation for analysis
he stops short of L(omega+2) because some of the structure provided by the
latter is "useless" (unbrauchbar). He surely would have changed his mind if
someone had pointed out that a good core result is provable in L(omega+2)
but not in his own system.

In sum: The early Weyl would not have accepted utility as a reason for
employing classical set theories. The early Weyl did accept utility as a
criterion for choosing between predicative set theories. The later Weyl
might have accepted utility as a reason for employing theories on the
borderline of evidentness.

Stephen Pollard
Professor of Philosophy

Division of Social Science
Truman State University
Kirksville, MO 63501

(660) 785-4653