[...] "point" has no meaning unless it is defined, and this involves

the specification of some sort of procedure. "All the points of a

line" as a purely intuitional concept apart from the rules by which

points are determined, can have no operational meaning, and

accordingly is to be held for mathematics as an entirely meaningless

concept. [...] "All the points of a line" means no more than "All the

rules for determining points on a line". [...] In other words, we have

no more reason to describe the points on a line as non-denumerable

than the non-terminating decimals. The repudiation of the diagonal

Verfahren for the decimals at the same time removes all reason for

thinking the points on a line non-denumerable.

In fact, a consistent application of the operational criterion of

meaning appears to demand the complete discard of the notion of

infinities of different orders. We never have "actual" infinites [...]

but only rules for operation which are not self-terminating. How can

there be different sorts of non-self-terminatingness? At any stage in

the process the rule either permits us to go on and take the next step

or it does not [...] and that is all there is to it. [...] Mengenlehre

is similarly supposed to have established the existence of

transcendentals by showing that all algebraic numbers are denumerable.

This proof I would reject, holding that the mere act of assigning

operational meaning to the transcendentals of itself ensures that they

are denumerable. As a matter of fact, only a few transcendentals have

been established. Mengenlehre is powerless to show whether any given

number, such as e or pi, is transcendental or not, and the detailed

analysis necessary in any given case for establishing transcendence is

not avoided by Mengenlehre. From the operational view a transcendental

is determined by a program of procedure of some sort; Mengenlehre has

nothing to add to the situation. And this, as far as my elementary

reading goes, exhausts the contributions which Mengenlehre has made in

other fields.

P.W. Bridgman: "A physicist's second reaction to Mengenlehre", Scripta

Mathematica, Vol. II, 1934.

Gruß, WM

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